BOREL ASYMPTOTIC DIMENSION AND HYPERFINITE …bseward/Files/asymdim.pdfymptotic dimension and its applications to group theory, see [2]). In recent years, a related notion of dynamic

BOREL ASYMPTOTIC DIMENSION AND HYPERFINITE

EQUIVALENCE RELATIONS

CLINTON T. CONLEY, STEVE C. JACKSON, ANDREW S. MARKS,BRANDON M. SEWARD, AND ROBIN D. TUCKER-DROB

Abstract. A long standing open problem in the theory of hyperfinite equiv-alence relations asks if the orbit equivalence relation generated by a Borel

action of a countable amenable group is hyperfinite. In this paper we prove

that this question always has a positive answer when the acting group is poly-cyclic, and we obtain a positive answer for all free actions of a large class of

solvable groups including the Baumslag–Solitar group BS(1, 2) and the lamp-

lighter group Z2 o Z. This marks the first time that a group of exponentialvolume-growth has been verified to have this property. In obtaining this result

we introduce a new tool for studying Borel equivalence relations by extending

Gromov’s notion of asymptotic dimension to the Borel setting. We show thatcountable Borel equivalence relations of finite Borel asymptotic dimension are

hyperfinite, and more generally we prove under a mild compatibility assump-tion that increasing unions of such equivalence relations are hyperfinite. As

part of our main theorem, we prove for a large class of solvable groups that all

of their free Borel actions have finite Borel asymptotic dimension (and finitedynamic asymptotic dimension in the case of a continuous action on a zero-

dimensional space). We also provide applications to Borel chromatic numbers,

Borel and continuous Følner tilings, topological dynamics, and C∗-algebras.

1. Introduction

The asymptotic dimension asdim(X, ρ) of a metric space (X, ρ) is a large-scaleanalog of Lebesgue covering dimension. It was first introduced by Gromov in 1993as a quasi-isometry invariant of finitely generated groups [15] (for a survey of as-ymptotic dimension and its applications to group theory, see [2]). In recent years,a related notion of dynamic asymptotic dimension was introduced in the contextof topological dynamical systems by Guentner, Willet, and Yu [17]. While a few ofour results pertain to topological dynamics and dynamic asymptotic dimension, ourmain focus is to further adapt the notion of asymptotic dimension to the contextof Borel equivalence relations, and to investigate its applications.

Let X be a standard Borel space, let ρ : X ×X → [0,+∞] be a Borel extendedmetric on X, where by extended we mean that the value +∞ is allowed, and definethe Borel equivalence relation Eρ = {(x, y) ∈ X × X : ρ(x, y) < ∞}. Whenevery class of Eρ is countable, we define the Borel asymptotic dimension of (X, ρ),denoted asdimB(X, ρ), to be d if d ∈ N ∪ {+∞} is least with the property that forevery radius r > 0 there exists a Borel equivalence relation F ⊆ Eρ such that theF -classes have uniformly bounded finite diameter and every ball of radius r meetsat most d + 1 many classes of F . The original notion of asymptotic dimension,

This paper was produced as the result of a SQuaRE workshop at the American Institute ofMathematics.

1

2 CONLEY, JACKSON, MARKS, SEWARD, AND TUCKER-DROB

which we will refer to as standard asymptotic dimension for clarity, is defined inthe same manner but without the Borel measurability restrictions. In particular,we always have asdimB(X, ρ) ≥ asdim(X, ρ).

It is perhaps a bit surprising that in the Borel context the most pertinent infor-mation is whether or not the Borel asymptotic dimension is finite, as the followingtheorem reveals. For the statement below, recall that a metric ρ is proper if everyball of finite radius has finite cardinality.

Theorem 1.1. Let X be a standard Borel space and let ρ be a Borel proper extendedmetric on X. If the Borel asymptotic dimension of (X, ρ) is finite, then it is equalto the standard asymptotic dimension of (X, ρ).

We also find that Borel and standard asymptotic dimension agree modulo ameager set.

Theorem 1.2. Let X be a Polish space and let ρ be a Borel proper extendedmetric on X. Then there is an Eρ-invariant dense Gδ set X ′ ⊆ X such thatasdimB(X ′, ρ) = asdim(X ′, ρ).

In this paper, our primary focus will be the Borel asymptotic dimensions as-sociated with Borel group actions and, ultimately, the subsequent applications tothe theory of hyperfinite equivalence relations. If G is a countable group andτ : G × G → [0,+∞) is a proper right-invariant metric on G, then to every Borelaction Gy X of G on a standard Borel space X one can associate a Borel properextended metric ρτ : X ×X → [0,+∞] by declaring ρτ (x, y) to be the minimum of{τ(1G, g) : g ∈ G, g · x = y} when this set is non-empty, and ∞ otherwise.

For a countable group G, the standard asymptotic dimension of (G, τ) does notdepend upon the choice of the proper right-invariant metric τ [2, Prop. 62] and isreferred to as the asymptotic dimension of G. Similarly, for a Borel action Gy Xthe Borel asymptotic dimension of (X, ρτ ) does not depend on the choice of τ(see Lemma 2.1). To simplify terminology, we will therefore speak of the Borelasymptotic dimension of the action Gy X and write asdimB(Gy X).

Our main theorem is below. Recall that a normal series for a group G is asequence G = G0 �G1 � . . . �Gn = {1G} of normal subgroups of G. We refer toGk/Gk+1 as a quotient of consecutive terms, and we call G0/G1 the top quotient.The standard asymptotic dimension of G is always bounded by the sum of thestandard asymptotic dimensions of the consecutive quotients in a normal series(see Thm. 68 in [2] and the remark following it).

Theorem 1.3. Let G be a countable group admitting a normal series where eachquotient of consecutive terms is a finite group or a torsion-free abelian group withfinite Q-rank, except the top quotient can be any group of uniform local polynomialvolume-growth or the lamplighter group Z2 o Z. If X is a standard Borel space andG y X is a free Borel action, then asdimB(G y X) = asdim(G) < ∞. Addition-ally, if X is a locally compact 0-dimensional second countable Hausdorff space andG acts freely and continuously on X, then the dynamic asymptotic dimension ofGy X is at most asdim(G) <∞.

The above theorem clearly applies to the lamplighter group, and it applies to allpolycyclic groups (i.e. groups admitting a normal series where every quotient ofconsecutive terms is a finitely-generated abelian group) and to many other solvablegroups as well. For instance, it applies to the Baumslag–Solitar group BS(1, 2) since

BOREL ASYMPTOTIC DIMENSION AND HYPERFINITE EQUIVALENCE RELATIONS 3

it is an extension of the dyadic rationals (a torsion-free abelian group of Q-rank 1)by Z.

Theorem 1.3 and the theorem below provide many new examples, both in theBorel and the topological settings, of groups having the property that all of theirfree actions admit Følner tilings. In view of the assumption of the theorem below,we remark that both Borel and dynamic asymptotic dimension are monotone de-creasing upon restricting an action to the action of a subgroup (see for instanceCorollary 2.2). In particular, the assumption of the next theorem will be met pro-vided G satisfies (or all of its finitely generated subgroups satisfy) the assumptionof Theorem 1.3.

Theorem 1.4. Let G be a countable amenable group, let X be a standard Borelspace, and let G y X be a free Borel action. Assume that asdimB(H y X) <∞ for every finitely generated subgroup H ≤ G. Then for every finite K ⊆ Gand δ > 0 there exist (K, δ)-invariant finite sets F1, . . . , Fn ⊆ G and Borel setsC1, . . . , Cn ⊆ X such that the map θ :

⊔ni=1 Fi × Ci → X given by θ(f, c) = f · c

is a bijection. Similarly, if X is a compact 0-dimensional Hausdorff space, G actscontinuously and freely on X, and the dynamic asymptotic dimension of H y X isfinite for every finitely generated subgroup H ≤ G, then the same conclusion holdswith the sets C1, . . . , Cn being additionally clopen.

The clopen Følner tilings provided by Theorems 1.3 and 1.4 immediately yieldinteresting consequences to topological dynamics and C∗-algebras, most notablyproviding new examples of classifiable crossed products.

Corollary 1.5. Suppose that G satisfies the assumption of Theorem 1.3, or thatall of its finitely generated subgroups do. Then every free continuous action of G ona compact metrizable space having finite covering dimension is almost finite. As aconsequence, the crossed products arising from minimal such actions are classifiedby the Elliot invariant (ordered K-theory paired with tracial states) and are simpleASH algebras of topological dimension at most 2.

Finite Borel asymptotic dimension also provides a bound on the Borel chromaticnumber χB(Γ) of a Borel graph Γ in terms of its standard chromatic number χ(Γ).The theorem below follows a theme of similar results. Specifically, Conley and Millerestablished the same bound for Baire-measurable chromatic numbers of locallyfinite Borel graphs, and when the graph is furthermore hyperfinite they obtainedthis bound for µ-measurable chromatic numbers [6]. More recently, Gao–Jackson–Krohne–Seward obtained this bound for the Borel chromatic numbers of Schreiergraphs induced by Borel actions of finitely-generated nilpotent groups [14].

Theorem 1.6. Let X be a standard Borel space, let Γ be a Borel graph on X, and letρ be the graph metric on Γ. Assume that Γ is locally finite and that asdimB(X, ρ) <∞. Then χB(Γ) ≤ 2 · χ(Γ)− 1.

In fact we obtain a result more general than the above by incorporating a newquantity we define called the asymptotic separation index.

From the above theorem and previous work of the authors [4], it follows thatthere are hyperfinite bounded degree acyclic Borel graphs having infinite Borelasymptotic dimension. This is in contrast to the fact that acyclic graphs alwayshave standard asymptotic dimension at most 1. On the other hand, in Lemma 8.3


we prove that the Borel asymptotic dimension of a Borel graph is at most 1 whenthe graph is induced by a bounded-to-one Borel function.

Lastly, we discuss applications to the theory of hyperfinite equivalence relations,which was our primary motivation in this work. Recall that an equivalence relationE on a standard Borel space X is Borel if E is a Borel subset of X ×X, and finite(or countable) if every E-class is finite (respectively countable). E is hyperfinite ifit is the union of an increasing sequence of finite Borel equivalence relations. Fromeach Borel action G y X of a countable group G we obtain the orbit equivalencerelation EXG = {(x, y) ∈ X × X : ∃g ∈ G g · x = y}, which is a countable Borelequivalence relation.

The study of Borel actions of countable groups is of fundamental significance tothe theory of countable Borel equivalence relations, as every countable Borel equiv-alence relation can be represented as the orbit equivalence relation of a Borel actionof a countable group [10]. Additionally, under the hierarchy of Borel reducibility,the hyperfinite equivalence relations are the simplest class of countable Borel equiv-alence relations whose study is non-trivial (as made precise by the Glimm-Effrosdichotomy and the equivalence of hyperfiniteness and Borel reducibility to E0; see[7, Thm. 7.1]). Thus, a fundamental problem is to determine for which actionsG y X is the orbit equivalence relation EXG hyperfinite. A long-standing andwidely-known formulation of this problem is the following.

Weiss’ Question [31]. If G is a countable amenable group, X a standard Borelspace, and G y X a Borel action, must the orbit equivalence relation EXG be hy-perfinite?

Every countable non-amenable group G admits a Borel action Gy X for whichEXG is not hyperfinite [19]. Thus, if Weiss’ question has a positive answer, then itprovides a characterization for the class of amenable groups.

In a striking result in 1980, Ornstein and Weiss proved that for every Borelaction G y X of a countable amenable group G the orbit equivalence relationEXG is µ-almost-everywhere hyperfinite for every Borel probability measure µ onX, meaning for each µ there is a G-invariant µ-conull set Y ⊆ X such that therestriction of EXG to Y is hyperfinite [25]. It was in light of this result that Weissposed his question in [31] and proceeded, in that same paper, to provide a positiveanswer to his question for Z (a second proof for Z was published by Slaman andSteel in 1988 [28]).

Since the conception of Weiss’ question, significant attention has gone to theproject of expanding the class of groups known to have a positive answer. In un-published work shortly after posing his question, Weiss proved the answer is positivefor the groups Zn. Years later, in 2002, Jackson, Kechris, and Louveau generalizedthis result by proving that EXG is hyperfinite whenever G is a finitely-generatedgroup of polynomial volume-growth [19, Thm. 1.16] (equivalently, whenever G hasa finite-index nilpotent subgroup [16, 32]). In an extremely technical and long proofthat introduced important new methods, Gao and Jackson obtained a positive an-swer for all countable abelian groups in 2007 [11]. Through a further expansionof those methods, a positive answer was obtained for all locally nilpotent groupsby Schneider and Seward in 2015 [27] (a group is locally nilpotent if every finitely-generated subgroup is nilpotent).

These last two results (of Gao–Jackson and Schneider–Seward) can be equiv-alently described as partially removing the finite-generation assumption from the


theorem of Jackson–Kechris–Louveau. Indeed, the classes of groups (abelian andlocally nilpotent) considered in these two works have the property that all theirfinitely-generated subgroups have polynomial volume-growth. The assumption ofpolynomial volume-growth is a universal trait of prior solutions to Weiss’ questionand is fundamental to the structure of all prior proofs. For this reason, relianceupon polynomial volume-growth has widely been viewed as a critical obstructionto further progress.

Through Theorem 1.3 we obtain the first solution to Weiss’ question that includesgroups of exponential volume-growth (any solvable group not containing a finite-index nilpotent subgroup has exponential volume-growth [32]). We obtain thisresult by relating Borel asymptotic dimension to hyperfiniteness.

Theorem 1.7. Let X be a standard Borel space and let ρ be a Borel proper extendedmetric on X. If the Borel asymptotic dimension of (X, ρ) is finite, then the finite-distance equivalence relation Eρ = {(x, y) ∈ X ×X : ρ(x, y) <∞} is hyperfinite.

Corollary 1.8. If G is a countable group satisfying the assumption of Theorem1.3, X is a standard Borel space, and Gy X is a free Borel action, then the orbitequivalence relation EXG is hyperfinite.

In the case of polycyclic groups, work of Schneider and Seward ([27, Cor. 8.2])allows us to remove the assumption that the actions are free. We thus obtain apositive answer to Weiss’ question for all polycyclic groups.

Corollary 1.9. If G is a polycyclic group, X is a standard Borel space, and Gy Xis any Borel action, then the orbit equivalence relation EXG is hyperfinite.

As previously mentioned, the works of Gao–Jackson [11] and Schneider–Seward[27] saw a tremendous increase in the length and technicality of their argumentsonly to partially remove the finite-generation assumption in the work of Jackson–Kechris–Louveau [19]. A compelling reason for why this occurred is that removingfinite-generation is equivalent to confronting a special case of another long-standingfundamental open problem on hyperfinite equivalence relations, one that is generallyconsidered harder than Weiss’ question. Specifically, the Union Problem asks: ifX is a standard Borel space and (En) is an increasing sequence of hyperfiniteequivalence relations on X, must the equivalence relation

⋃nEn be hyperfinite? If

we express a group G as an increasing union of finitely generated subgroups Gn,then the overlap between this problem and Weiss’ question can be seen by observingthat for every action Gy X we have EXG =

⋃nE

XGn

.Using the notion of Borel asymptotic dimension, we solve a special case of the

Union Problem.

Theorem 1.10. Let X be a standard Borel space and let (ρn)n∈N be a sequence ofBorel proper extended metrics on X. Assume that for every n ∈ N and r > 0 theρn-balls of radius r have uniformly bounded ρn+1-diameter, and set E =

⋃nEρn .

If (X, ρn) has finite Borel asymptotic dimension for every n then E is hyperfinite.

The above theorem allows us to immediately expand on Corollary 1.8.

Corollary 1.11. If G is the union of groups satisfying the assumption of Theorem1.3, X is a standard Borel space, and Gy X is a free Borel action, then the orbitequivalence relation EXG is hyperfinite.


Theorem 1.10 also allows us to, for the first time, fully remove the finite-generationassumption from the work of Jackson–Kechris–Louveau [19]. Indeed, it is simpleto prove (as we do in Corollary 5.5) that Borel actions Gy X have finite Borel as-ymptotic dimension whenever G is a finitely-generated group of polynomial volume-growth.

Corollary 1.12. Let G be a countable group with the property that every finitelygenerated subgroup of G has polynomial volume-growth (equivalently, every finitelygenerated subgroup of G contains a finite-index nilpotent subgroup). If X is astandard Borel space and G y X is any Borel action then the orbit equivalencerelation EXG is hyperfinite.

For instance, a new group to which the above corollary applies but prior resultsdo not is the group (

⊕n∈N Z) o S∞, where S∞ is the group of finitely-supported

permutations of N.In addition to the formal results we obtain, we believe that the methods we

employ are equally valuable. The introduction of Borel asymptotic dimension intothe study of hyperfinite equivalence relations is exciting both for the new resultsit leads to, as discussed above, and for how it reshapes our understanding of priorresults. Specifically, the papers by Gao–Jackson [11] and Schneider–Seward [27],which were primarily devoted to partially removing the finite-generation assumptionin the work of Jackson–Kechris–Louveau [19], were quite long (74 and 45 pages,respectively) and highly technical. Theorem 1.10 achieves the chief goal of thosepapers, but its proof is less technical and tremendously shorter (only a few pages,including all supporting lemmas). Together with Corollary 5.5 (which also hasa very short proof), these self-contained arguments encompass all prior work onWeiss’ question and a bit more. Our proofs certainly differ significantly in terms offormal concepts and details, however we find it both surprising and satisfying that,in our opinion, our arguments capture the core intuitive components of those priorworks. We hope therefore that this work can lead to a renewed understanding ofprior discoveries and can reinvigorate the pursuit of Weiss’ question.

Outline. Section 2 contains a review of preliminary material. In Section 3 weintroduce Borel asymptotic dimension and asymptotic separation index. Many ofthe remaining sections are mostly independent of one another. In Section 4 weprove that Borel asymptotic dimension is equal to standard asymptotic dimensionwhen the former is finite, and we prove that in many situations the asymptoticseparation index is at most 1. In the first half of Section 5 we retrace part ofthe work of Jackson–Kechris–Louveau, but in the language of Borel asymptoticdimension, and prove that Borel actions of groups of polynomial volume-growthhave finite Borel asymptotic dimension. In the second half of Section 5 we studypacking phenomena for certain group extensions and, relying upon that, in Section6 we prove our main theorem that all free actions of groups with suitable normalseries have finite Borel asymptotic dimension. In section 7 we prove that finiteBorel asymptotic dimension implies hyperfiniteness and, relying a bit on Section4, we use Borel asymptotic dimension to solve an instance of the Union Problem.Finally, we study applications to Borel chromatic numbers in Section 8, Følnertilings in Section 9, and topological dynamics and C∗-algebras in Section 10. Muchof this last section discusses alterations to proofs from earlier in the paper, arguinghow clopen sets can be used in place of Borel sets.


2. Preliminaries

As is customary in logic, throughout the paper we identify each positive integern with the set {0, 1, . . . , n− 1}.

By an extended metric we mean a metric that is allowed to take value +∞.Given an extended metric ρ on X we write Eρ = {(x, y) ∈ X ×X : ρ(x, y) < ∞}for the finite-distance equivalence relation. We let B(x; r) denote the closed r-ballcentered at x, and we call ρ proper if every ball of finite radius has finite cardinality.

2.1. Groups and group actions. A normal series for a group G is a sequenceG = G0 �G1 � . . .�Gn = {1G} of normal subgroups of G. We refer to Gk/Gk+1

as a quotient of consecutive terms, and we call G0/G1 the top quotient.If G is an abelian group, then its torsion subgroup is the subgroup consisting of

all elements of finite order. G is called torsion-free if every non-identity element inG has infinite order. The Q-rank of G (when G is written additively) is the largestcardinality of a Z-linearly independent subset, or equivalently the dimension of theQ-vector space Q⊗G.

If P is a property of groups, then we will say that G is locally P if every finitelygenerated subgroup of G has property P , and we will say G is virtually P if Gcontains a finite-index subgroup with property P .

A finitely generated group G has polynomial volume-growth if there is d ∈ N(the least of which is called the degree) so that for any (equivalently every) finitesymmetric generating set B containing 1G there is c > 0 with |Br| ≤ crd for allr ≥ 1. By Gromov’s theorem [16, 32], this is equivalent to G being finitely generatedand virtually nilpotent.

We say that G has local polynomial volume-growth if every finitely generatedsubgroup of G has polynomial volume-growth, and has uniform local polynomialvolume-growth if there is d ∈ N (the least of which is referred to as the degree)so that every finitely generated subgroup of G has polynomial volume-growth ofdegree at most d. For instance, locally finite groups have uniform local polynomialvolume-growth of degree 0 and countable abelian groups of Q-rank d have uniformlocal polynomial volume-growth of degree d.

Every finitely generated group G can naturally be viewed as a metric space.Specifically, to any finite symmetric generating set A containing 1G one can asso-ciate the A-word length metric τA defined by letting τA(g, h) be the least n ∈ Nwith gh−1 ∈ An. Then τA is a proper right-invariant metric. More generally, everycountable group G admits some proper right-invariant metric. For instance, if Gis finite we can let τ be the discrete metric, and otherwise we can fix a bijectionw : Z+ → G and define

τ(g, h) = min{n1 + n2 + . . .+ nk : k ≥ 1, w(n1)w(n2) · · ·w(nk) = gh−1}.

An action G y X is free if g · x 6= x for every x ∈ X and non-identity g ∈ G.If G acts on X and τ is a proper right-invariant metric on G, then we can definea proper extended metric ρτ on X by letting ρτ (x, y) be the minimum of τ(1G, g)over all g ∈ G satisfying g · x = y, and be +∞ if no such g exists. Since τ isright-invariant, for any fixed x ∈ X the map g 7→ g · x from (G, τ) to (X, ρτ ) is1-Lipschitz and is an isometry when the action is free.

2.2. Standard asymptotic dimension. Let (X, ρ) be an extended metric space.We say that an equivalence relation E on X is bounded if every E-class has finite


Ball of radiusr = 4r0

Figure 1. An equivalence relation E satisfying condition (1) whenX = R2 and d = 2.

ρ-diameter, and is uniformly bounded if there is a uniform finite bound to the ρ-diameters of E-classes. We also say E is countable if every E-class is countable,and finite if every E-class is finite. Moreover, E is uniformly finite if there is afinite uniform bound to the cardinalities of the E-classes.

For U ⊆ X and r > 0, we denote by Fr(U) the smallest equivalence relationon U with the property that (x, y) ∈ Fr(U) whenever x, y ∈ U satisfy ρ(x, y) ≤ r.Similarly, for an action G y X, U ⊆ X, and finite A ⊆ G, we let FA(U) be thesmallest equivalence relation on U satisfying (x, y) ∈ FA(U) whenever x, y ∈ U andx ∈ A · y or y ∈ A · x.

There are several equivalent definitions for the standard asymptotic dimension ofa metric space (X, ρ) (see for instance [2, Thm. 19]), but we will only be interestedin two. Consider the following statements.

(1) For every r > 0 there is a uniformly bounded equivalence relation E on Xsuch that for every x ∈ X the ball B(x; r) meets at most d+ 1 classes of E.

(2) For every r > 0 there are sets U0, . . . , Ud that cover X and have the propertythat Fr(Ui) is uniformly bounded for every i.

Then (1) and (2) are equivalent for every d ∈ N (this is a standard fact, recorded forinstance in [2, Thm. 19] and reproved in the Borel setting in the next section). Thestandard asymptotic dimension of (X, ρ), denoted asdim(X, ρ), is the least d ∈ Nfor which (1) and (2) are true, or +∞ if no such d exists. We will use this samedefinition for extended metric spaces as well.

A visual demonstration, based on (1), that R2 has standard asymptotic dimen-sion 2 is depicted in Figure 1. In that figure, the classes of E are rectangles ofside-length greater than 4r, and the rectangles are staggered so that every r-ballmeets at most 3 rectangles, rather than 4. A similar demonstration, based on (2),is depicted in Figure 2. The objects depicted in these figures are connected by aconstruction that will be discussed in the next section.

We define the standard asymptotic dimension of a group G, denoted asdim(G),and the standard asymptotic dimension of an action G y X, denoted asdim(G yX), to be asdim(G, τ) and asdim(X, ρτ ), respectively, for any proper right-invariantmetric τ on G. These two quantities do not depend on the choice of τ .

Lemma 2.1. Let G y X be an action of a countable group G. Then asdim(G yX) is equal to the least d ∈ N with the property that for every finite set A ⊆ Gthere are sets U0, . . . , Ud covering X such that FA(Ui) is uniformly finite for everyi ∈ d+ 1, and is equal to ∞ if no such d exists.


Ball of radiusr0

Figure 2. Sets U0 (pink), U1 (green), U2 (blue) satisfying condi-tion (2) when r = r0, X = R2, and d = 2.

Proof. This follows from the basic fact that if τ is any proper right-invariant metricon G then for any r ≥ 0 the set A = {a ∈ G : τ(a, 1G) ≤ r} is finite and for anyfinite A ⊆ G there is an r ≥ 0 with τ(a, 1G) ≤ r for all a ∈ A. We leave it as aneasy exercise for the reader to fill in the details. �

We note the following immediate consequence.

Corollary 2.2. Let G y X be an action of a countable group G. If H ≤ Gthen the restricted action satisfies asdim(H y X) ≤ asdim(G y X). Moreover,asdim(G y X) is the supremum of asdim(H y X) as H varies over all finitelygenerated subgroups of G.

Similar statements hold for the standard asymptotic dimension of groups. Togive a few examples, locally finite groups have standard asymptotic dimension 0,Qd has standard asymptotic dimension d, solvable groups have standard asymp-totic dimension bounded by their Hirsch length (i.e. the sums of the Q-ranks ofthe abelian quotients appearing in their derived series, which may be infinite), andnon-abelian free groups have standard asymptotic dimension 1. On the other hand,⊕

n∈N Z, Z oZ, and the Grigorchuk group have infinite standard asymptotic dimen-sion, as does any group having them as a subgroup. Further examples can be foundin [2]. It is also worth noting that every countable group has standard asymptoticdimension bounded by the sum of the standard asymptotic dimensions of the quo-tients of consecutive terms in any normal series (see Thm. 68 in [2] and the remarkfollowing it).

2.3. Borel equivalence relations and graphs. A set X equipped with a distin-guished σ-algebra B(X) is called a standard Borel space if there exists a separableand completely metrizable topology on X such that B(X) coincides with the σ-algebra generated by the open sets. In this setting, we call the members of B(X)Borel sets. When X is a standard Borel space and ρ is a Borel measurable extendedmetric on X, we call (X, ρ) a Borel extended metric space.

An equivalence relation E on X is Borel if E is a Borel subset of X×X. A Borelequivalence relation E is hyperfinite if it can be expressed as the increasing unionof a sequence of finite Borel equivalence relations En. For x ∈ X we write [x]E forthe E-class of x, and for Y ⊆ X we write [Y ]E = {x ∈ X : ∃y ∈ Y (x, y) ∈ E} forthe E-saturation of Y .


The equivalence relations we consider will always be countable. An importantfeature of countable Borel equivalence relations is that the Lusin-Novikov uni-formization theorem provides Borel functions fn : X → X with the property thatE = {(x, fn(x)) : x ∈ X, n ∈ N} [20, Thm. 18.10]. This fact is quite useful forverifying that sets are Borel. For instance, if E is a countable Borel equivalencerelation and Y ⊆ X is Borel, then [Y ]E is Borel as well because if (fn)n∈N is asdescribed above then x ∈ [Y ]E if and only if there is n ∈ N with fn(x) ∈ Y . Thenext lemma provides another example.

Lemma 2.3. Let X be a standard Borel space and let U ⊆ X be Borel.

(1) If ρ is a Borel extended metric on X and Eρ is countable, then Fr(U) isBorel for every r ≥ 0.

(2) If Gy X is a Borel action then FA(U) is Borel for every finite A ⊆ G.

Proof. Consider (a). Since Eρ is countable, we may let fn : X → X, n ∈ N,be as given by the Lusin–Novikov uniformization theorem. Then (x, y) ∈ Fr(U)if and only if there is k ∈ N and n0, . . . , nk ∈ N with fn0

(x) = x, fnk(x) = y,

ρ(fni(x), fni+1

(x)) ≤ r for all i < k, and fni(x) ∈ U for all i ≤ k. Thus Fr(U)

is Borel. The proof of (b) is nearly identical, but one can use the elements of Ginstead of the fn’s. �

In our arguments the equivalence relations Fr(U) and FA(U) will often be finite.A useful feature of finite Borel equivalence relations is that they admit Borel selec-tors, meaning there is a Borel function s : X → X satisfying s(y) = s(x) E x for all(x, y) ∈ E [20, Thm. 12.16]. In particular, since s is finite-to-one the image s(X)is a Borel set that contains exactly one point from every E-class [20, Ex. 18.14].

If X is a standard Borel space, a Borel graph on X is a set of edges Γ that isa Borel subset of X × X. The graph metric ρ is defined by letting ρ(x, y) be theshortest possible length of a path joining x and y, or∞ if no such path exists. Thismetric is proper when Γ is locally finite, meaning every x ∈ X has finite degree inΓ. A set Y ⊆ X is Γ-independent if no two points in Y are joined by an edge, anda proper coloring of Γ is a function c : X → Z with the property that c−1(z) isΓ-independent for every z ∈ Z. The chromatic number of Γ, denoted χ(Γ), is theleast cardinality of a set Z such that there exists proper coloring c : X → Z. WhenX is a standard Borel space and Γ is a Borel graph, the Borel chromatic numberχB(Γ) is defined in the same way but with the requirement that Z be a standardBorel space and that c be Borel measurable.

The following proposition gives a case when the Borel chromatic number is equalto the standard chromatic number:

Proposition 2.4 ([6, Proposition 1]). Suppose that X is a standard Borel spaceand Γ is a locally countable Borel graph on X whose connectedness relation EΓ onX admits a Borel selector. Then χ(G) = χB(G).

We will use this proposition in the case when the Borel graph Γ has finite con-nected components and hence has a Borel selector by the above. In this specialcase, one way of proving this proposition is by taking a Borel linear ordering ofX and then taking the lexicographically least coloring using χ(G)-colors in eachconnected component. It is not hard to verify this coloring is Borel.


3. Borel Asymptotic Dimension

In this section we introduce Borel asymptotic dimension. We first translate intothe Borel setting the two conditions discussed in Section 2 that define standardasymptotic dimension, and we check that they are equivalent when Eρ is countable.

Lemma 3.1. Let (X, ρ) be a Borel extended metric space with Eρ countable. Thenfor every d ∈ N the following statements are equivalent.

(1) For every r > 0 there is a uniformly bounded Borel equivalence relation Esuch that for every x ∈ X the ball B(x; r) meets at most d+ 1 classes of E.

(2) For every r > 0 there are Borel sets U0, U1, . . . , Ud that cover X and havethe property that Fr(Ui) is uniformly bounded for every i ∈ d+ 1.

Furthermore, if (1’) and (2’) are the statements where “uniformly bounded” is re-placed by “bounded” in (1) and (2) respectively, then we similarly have that (1’)and (2’) are equivalent for every d ∈ N.

Proof. Since Eρ is countable, the Lusin–Novikov uniformization theorem providesBorel functions fn : X → X, n ∈ N, with Eρ = {(x, fn(x)) : x ∈ X, n ∈ N}. Weprove that (1) and (2) are equivalent; the equivalence of (1’) and (2’) follows thesame argument but replacing “uniformly bounded” with “bounded” throughout.

(1) ⇒ (2). Fix r0 > 0. Apply (1) with r = (d + 2)r0 to obtain a uniformlybounded Borel equivalence relation E such that for every x ∈ X the (d+ 2)r0-ballaround x meets at most d + 1 classes of E. For each 0 ≤ i ≤ d define Ui to bethe set of x ∈ X having the property that both B(x; (i+ 1)r0) and B(x; (i+ 2)r0)meet i+ 1 many classes of E. Figure 2 depicts the sets U0, U1, U2 that are obtainedwhen this construction is applied to the relation E in Figure 1. Notice that B(x; r0)must meet at least one E-class and, by our choice of E, B(x; (d+ 2)r0) must meetstrictly less than d + 2 many E-classes. Thus monotonicity properties imply that{Ui : 0 ≤ i ≤ d} covers X. Additionally, each Ui is Borel because for every i, jthe set Ci,j = {x ∈ X : B(x; (i + 1)r0) meets at least j classes of E} is Borel, asx ∈ Ci,j if and only if there are n1, . . . , nj ∈ N with (fns

(x), fnt(x)) 6∈ E and

ρ(x, fns(x)) ≤ (i+ 1)r0 for all s 6= t.Now suppose that x, x′ ∈ Ui and ρ(x, x′) ≤ r0. We claim that for every class

D ∈ X/E we have that B(x; (i+ 1)r0) meets D if and only if B(x′; (i+ 1)r0) meetsD. Indeed, if B(x; (i+1)r0) meets D then by the triangle inequality B(x′; (i+2)r0)meets D, and since B(x′; (i + 1)r0) and B(x′; (i + 2)r0) meet the same number ofE-classes, we conclude that B(x′; (i+1)r0) meets D. From this claim it follows thatfor every x ∈ Ui the Fr0(Ui)-class of x is contained in the ball of radius (i + 1)r0

around [x]E . So Fr0(Ui) is uniformly bounded since E is uniformly bounded.(2) ⇒ (1). Fix r0 > 0. Apply (2) with r = 2r0 to get Borel sets U0, U1, . . . , Ud.

By replacing each Ui with Ui \⋃j<i Uj we can assume that the Ui’s partition X.

Then E =⋃i F2r0(Ui) is a uniformly bounded Borel equivalence relation on X. For

every x and every i we have that B(x; r0) can meet at most one class of F2r0(Ui),and thus B(x; r0) meets at most d+ 1 many classes of E. �

Since the non-Borel versions of (1) and (2) are both commonly used to definestandard asymptotic dimension and these conditions remain equivalent in the Borelsetting when Eρ is countable, we restrict our definition of Borel asymptotic dimen-sion to this setting. We leave open how to define Borel asymptotic dimension inother situations.


Definition 3.2. Let (X, ρ) be a Borel extended metric space such that Eρ iscountable. The Borel asymptotic dimension of (X, ρ), denoted asdimB(X, ρ) is theleast d ∈ N for which the statements (1) and (2) in Lemma 3.1 are true, and is ∞if no such d exists. Similarly, the asymptotic separation index of (X, ρ), denotedasi(X, ρ) is the least d ∈ N for which the statements (1’) and (2’) are true, and is∞ if no such d exists.

The notion of asymptotic separation index is a new concept that we introduce.We will see that this notion carries useful consequences, however it is unclear howrich this notion is. For one, the notion is useless when Borel constraints are removedas it is always at most 1 (pick a single point from each Eρ-class and use annuli ofinner and outer radii 2rn and 2r(n+ 1), n ∈ N, to determine the classes of E as in(1’)). Moreover, we will prove in the next section that the asymptotic separationindex is often at most 1, and we know of no example where it is both finite andgreater than 1.

For a Borel action G y X of a countable group G, we can define the Borelasymptotic dimension asdimB(G y X) to be asdimB(X, ρτ ) for any choice ofproper right-invariant metric τ on G. Once again this quantity does not dependon the choice of τ and the condition in Lemma 2.1 again characterizes the valueasdimB(G y X) when the sets Ui are required to be Borel. Similarly, Borelasymptotic dimensions of group actions have the same monotonicity propertiesdescribed in Corollary 2.2. Similar statements apply to the asymptotic separationindex of a Borel action Gy X, but we will not need this.

4. General properties

In this section we establish some basic constraints on the values of Borel asymp-totic dimension and asymptotic separation index. We first note that these valuescan be 0 only in fairly trivial situations.

Corollary 4.1. Let (X, ρ) be a Borel extended metric space with Eρ countable.Then asdimB(X, ρ) = 0 if and only if Fr(X) is uniformly bounded for every r > 0.Similarly, asi(X, ρ) = 0 if and only if Fr(X) is bounded for every r > 0.

Proof. This is immediate from conditions (2) and (2’) of Lemma 3.1, as we musthave U0 = X. �

We next prove that Borel asymptotic dimension can only differ from its standardcounterpart when it is infinite.

Theorem 4.2. Suppose that (X, ρ) is a proper Borel extended metric space. Ifasi(X, ρ) <∞ (or, less generally, if asdimB(X, ρ) <∞), then the Borel asymptoticdimension and the standard asymptotic dimension of (X, ρ) are equal.

To prove this we need two supporting lemmas. We say that the standard (re-spectively Borel) (r,R)-dimension of (X, ρ) is d if d ∈ N is least with the propertythat there exist sets (respectively Borel sets) U0, . . . , Ud covering X such that forevery i each class of Fr(Ui) has diameter at most R, and is ∞ if no such d exists.

Lemma 4.3. Let (X, ρ) be a proper Borel extended metric space having standard(r,R)-dimension d. If Y ⊆ X is Borel and Fr(Y ) is bounded, then Y has Borel(r,R)-dimension at most d.


Proof. This lemma is proved using a similar idea to Proposition 2.4. We assumed < ∞, as otherwise there is nothing to prove. Since ρ is proper, Fr(Y ) is a finiteBorel equivalence relation, and each equivalence class admits only finitely manypartitions into d+1 pieces. Let X [<∞] be the standard Borel space of finite subsetsof X. Let P be the set of (D0, . . . , Dd) ∈

∏i≤dX

[<∞] such that D0, . . . , Dd are

disjoint, and their union C = D0 ∪ . . . ∪Dd is an equivalence class of Fr(Y ). LetP ′ ⊆ P be the set of (D0, . . . , Dd) ∈ P such that for each i, the Fr(Di)-classeshave diameter at most R. Since we are assuming Y has (r,R)-diameter at mostd, each Fr(Y ) class C has at least one partition in P ′. It is easy to check usingLusin-Novikov uniformization that P and P ′ are Borel. We want to find a Borelway of taking each Fr(Y ) class C, and choosing exactly one partition of it that liesin P ′.

Let F be the equivalence relation on P ′ where (D0, . . . , Dd) F (D′0, . . . , D′d) if⋃

i≤dDi =⋃i≤dD

′i, so they partition the same Fr(Y ) class. Then F is a Borel

equivalence relation with finite classes, so we can use a Borel selector to obtain aBorel set P0 ⊆ P ′ that contains exactly one partition of each Fr(Y ) class. The setsUi = {x : (∃(D0, . . . , Dd) ∈ P0)x ∈ Di} are Borel by Lusin-Novikov uniformization,and they witness that Y has Borel (r,R)-dimension at most d. �

Lemma 4.4. Suppose that (X, ρ) is a Borel extended metric space, and that X isthe union of Borel sets X0∪· · ·∪Xn−1. Assume that each Xi has Borel (15ri, ri+1)-dimension at most d, where the ri’s are non-decreasing. Then (X, ρ) has Borel(r0, 5rn)-dimension at most d.

Proof. This is trivial if d = ∞, so we assume d < ∞. Passing to a subset can-not increase Borel asymptotic dimension, so we may assume that the sets Xi arepairwise disjoint. By induction on n, it suffices to show that if {X0, X1} is a Borelpartition of X where X0 has Borel (r0, r1)-dimension at most d and X1 has Borel(3r1, r2)-dimension at most d, then X has Borel (r0, 5r2)-dimension at most d.

Our assumptions imply that there are Borel sets U00 , . . . , U

0d covering X0 where

the diameter of every class in Fr0(U0i ) is at most r1, and there are Borel sets

U10 , . . . , U

1d covering X1 where the diameter of every class in F3r1(U1

i ) is at most

r2. Define Ui = U0i ∪ U1

i . Then the Ui’s are Borel and cover X. Notice that theclasses of Fr0(Ui) are obtained by gluing together classes of Fr0(U0

i ) with classes of

Fr0(U1i ) whenever the distance between them is at most r0. If D0 is a Fr0(U0

i )-class

then, since D has diameter at most r1 and 2r0 + r1 ≤ 3r1, the set of points in U1i

that are distance at most r0 from D0 must lie in a single F3r1(U1i )-class. Therefore,

every class of Fr0(Ui) is either a Fr0(U0i )-class or else it is the union of a subset B of

some F3r1(U1i )-class and the Fr0(U0

i )-classes that are distance at most r0 to B. Weconclude that every Fr0(Ui)-class has diameter at most r2 + 2(r0 + r1) ≤ 5r2. �

Proof of Theorem 4.2. Suppose that the standard asymptotic dimension of (X, ρ)is d and that its asymptotic separation index is s < ∞. Fix r0; we will find someR such that (X, ρ) has Borel (r0, R)-dimension d.

As (X, ρ) has standard asymptotic dimension d, there is some r1 ≥ r0 such thatX has standard (15r0, r1)-dimension at most d. Continue this process, choosing foreach i ∈ s + 1 some ri+1 ≥ ri such that X has standard (15ri, ri+1)-dimension atmost d. Finally, using the fact that (X, ρ) has asymptotic separation index s, finda Borel partition {X0, X1, . . . , Xs} of X such that Frs(Xi) is bounded for everyi ∈ s+1. Lemma 4.3 implies that each Xi has Borel (15ri, ri+1)-dimension at most


d, and thus Lemma 4.4 ensures that (X, ρ) has Borel (r0, 5rs+1)-dimension at mostd as required. �

We next turn our attention to asymptotic separation index and will need animportant technique that will recur throughout this paper. Asymptotic dimensionis fundamentally concerned with bounding the multiplicities of boundaries at anyprescribed large scale (as can be seen most clearly from condition (1) of Lemma3.1). The technique we need therefore relates to ensuring that boundaries stayseparated from each other, and is based on two simple observations. The first isthat if the classes of F2a(U1) have diameter at most b, then the sets B(D; a), whereD varies over all F2a(U1)-classes, are pairwise disjoint and have diameter at most2a + b. So for any given set V2, if we union V2 with the sets B(D; a) meeting V2,we obtain a set V2 ⊆ U2 ⊆ B(V2; 2a + b) with the property that for every x ∈ U1

the set B(x; a) is either disjoint with U2 or contained in U2 (i.e. the boundaries ofU1 and U2 are separated by distance at least a). The second is that if U ⊆ B(V ; a)then every Fr(U)-class is contained within distance a of some F2a+r(V )-class; inparticular, if F2a+r(V ) is uniformly bounded then so is Fr(U). The next lemmaformalizes this technique when applied to a sequence (Vn)n∈N.

Lemma 4.5. Let (X, ρ) be a Borel extended metric space with Eρ countable and foreach n ∈ N let Vn ⊆ X be Borel and let an, bn, rn ≥ 0. Assume for every n ∈ N thatevery F6an(Vn)-class has diameter at most bn and that an ≥ rn +

∑n−1k=0 4ak + bk.

Then there are Borel sets Un ⊇ Vn such that Frn(Un) is uniformly bounded forevery n and whenever n < m and x ∈ Un the set B(x; rn) is either disjoint with orcontained in Um.

Proof. Notice that every F2an(B(Vn; 2an))-class is contained within distance 2anof some F6an(Vn)-class and thus has diameter at most 4an + bn. For Y ⊆ Xdefine the saturation sn(Y ) = Y ∪ [Y ∩ B(Vn; 2an)]F2an

(B(Vn;2an)). Then sn(Y ) is

Borel when Y is Borel and sn(Y ) ⊆ B(Y ; 4an + bn). Set U0 = V0, for n ≥ 1 setUn = s0 ◦ · · · ◦ sn−1(Vn), and notice Un ⊆ B(Vn; an). From this last containmentand the inequality rn ≤ an, we see that Frn(Un) is uniformly bounded. Nowconsider n < m and x ∈ Un and assume that B(x; rn) is not a subset of Um. SinceB(x; an) ⊆ B(Vn; 2an) is contained in a single F2an(B(Vn; 2an))-class, it followsfrom the definition of sn and Um that B(x; an) is disjoint with W = sn ◦ · · · ◦sm−1(Vm). Since Um ⊆ B(W ;

∑n−1k=0 4ak + bk) and rn +

∑n−1k=0 4ak + bk ≤ an, it

follows that B(x; rn) is disjoint with Um. �

Below, for sets A and B, we say that A divides B if A∩B is a non-empty propersubset of B.

Corollary 4.6. Let (X, ρ) be a Borel extended metric space with Eρ countable andwith Borel asymptotic dimension d < ∞, and let (rn)n∈N be a sequence of radii.Then there exists a sequence of Borel covers Un = {Un0 , . . . , Und } of X satisfying:

(i) Frn(Uni ) is uniformly bounded for every n and i;(ii) if n < m, then for every x ∈ Uni the set B(x; rn) is either disjoint with or

a subset of Umi .

Proof. Set a0 = r0. Given an > 0, pick Borel sets V n0 , . . . , Vnd covering X with

the property that F6an(V ni ) is uniformly bounded for every i, let bn > 0 boundthe diameters of every class of each relation F6an(V ni ), and set an+1 = rn+1 +

BOREL ASYMPTOTIC DIMENSION AND HYPERFINITE EQUIVALENCE RELATIONS 15∑nk=0 4ak + bk. For every i ∈ d + 1, apply Lemma 4.5 to {V ni : n ∈ N} to obtain

Borel sets Uni ⊇ V ni satisfying conditions (i) and (ii). �

We will now prove that the asymptotic separation index is often at most 1 (infact we know of no examples where it is both finite and greater than one). Ourproof will use the following criterion.

Lemma 4.7. Let (X, ρ) be a Borel extended metric space with Eρ countable. As-sume that for every r > 0 there are Borel sets Vn ⊆ X, n ∈ N, such that F3r(Vn)is bounded for every n, for every x there is n with B(x; r) ⊆ Vn, and whenevern < m and x ∈ Vn the set B(x; 4r) is either contained in or disjoint with Vm. Thenasi(X, ρ) ≤ 1.

Proof. Let r > 0 and let (Vn)n∈N be as described. Let U1 be the set of x ∈ X forwhich there is an n with Vn dividing B(x; r), and set U0 = X \U1. For x ∈ U1 let nxbe such that Vnx

divides B(x; r), and for x ∈ U0 let nx be least with B(x; r) ⊆ Vnx.

Clearly if x, y ∈ U0 and ρ(x, y) ≤ r then nx = ny. Similarly, if x, y ∈ U1 andρ(x, y) ≤ r then B(y; 2r) is divided by both Vnx

and Vnyand hence ny = nx. So

for x ∈ Ui we have [x]Fr(Ui) is contained in a Fr(B(Vnx ; r))-class, and has finitediameter since F3r(Vnx

) is bounded. �

The above criterion has some rough similarity with the notions of “toast” in[12, 13, 14] and “barriers” in [6]. In fact, the proof of (b) below is similar to theproof of [6, Thm. B].

Theorem 4.8. Let (X, ρ) be a Borel extended metric space with Eρ countable.

(a) If asdimB(X, ρ) <∞ then asi(X, ρ) ≤ 1.(b) Assume ρ is proper and fix a compatible Polish topology on X. Then there

is an Eρ-invariant dense Gδ set X ′ ⊆ X such that asi(X ′, ρ) ≤ 1.

We point out that item (b) and Theorem 4.2 together imply Theorem 1.2 fromthe introduction.

Proof. (a). Set d = asdimB(X, ρ) and let r > 0. Set rn = 4r + n and applyCorollary 4.6 to obtain a sequence of Borel covers Un = {Un0 , . . . , Und } satisfyingclauses (i) and (ii) of that corollary. For each x ∈ X define c(x) ∈ d + 1 to beleast with x ∈ Unc(x) for infinitely many n. An immediate consequence of (ii) is that

B(x; rn) ⊆ Umc(x) for every n and infinitely many m ≥ n. Consequently, c(x) = c(y)

when ρ(x, y) <∞. Thus c is Borel and Eρ-invariant. Finally, it is easily seen thatthe sets Vn =

⋃i∈d+1 U

ni ∩ c−1(i) satisfy the conditions of Lemma 4.7.

(b). Let r > 0. Set a0 = 4r and inductively define an+1 = 4r+∑nk=0(4ak + 2r).

Let Γn be the Borel graph on X where (x, y) ∈ Γn if and only if 0 < ρ(x, y) ≤6an + 2r. Then Γn is locally finite since ρ is proper, so [21, Prop. 4.3] implies thatthere is a Borel proper coloring cn : X → N of Γn. Set Y ni = c−1

n (i).Consider the set P of pairs (x, s) ∈ X × NN with the property that for every

x′ with ρ(x, x′) < ∞ there is n with x′ ∈ Y ns(n). We claim there is a comeager set

of s ∈ NN such that {x ∈ X : (x, s) ∈ P} is comeager. By the Kuratowski–Ulamtheorem [20, Thm. 8.41] it suffices to check that for every x ∈ X the set {s ∈ NN :(x, s) ∈ P} is comeager. By the Lusin–Novikov uniformization theorem, there areBorel functions fk : X → X, k ∈ N, with Eρ = {(x, fk(x)) : x ∈ X, k ∈ N}. Sincefor every k and n there is i ∈ N with fk(x) ∈ Y ni , it is immediate that the set


{s ∈ NN : ∃n ∈ N fk(x) ∈ Y ns(n)} is open and dense. By taking the intersection

over k ∈ N, we find that the set {s ∈ NN : (x, s) ∈ P} is indeed comeager for everyx ∈ X.

Now fix s ∈ NN with the property that the set X ′ = {x ∈ X : (x, s) ∈ P} iscomeager. From the definition of P it is immediate that X ′ is Eρ-invariant. SetWn = B(Y ns(n); r) ∩ X

′. By construction each class of F6an(Wn) coincides with

B(y; r) for some y ∈ Y ns(n) and thus has diameter at most 2r. Therefore we can

apply Lemma 4.5, using rn = 4r for all n, to obtain Borel sets Vn ⊇ Wn such thatF3r(Vn) is uniformly bounded for every n and whenever n < m and x ∈ Vn the setB(x; 4r) is either disjoint with or contained in Vm. Additionally, by our choice of sfor every x ∈ X ′ there is n with x ∈ Y ns(n) and hence B(x; r) ⊆Wn ⊆ Vn. Therefore

asi(X ′, ρ) ≤ 1 by Lemma 4.7. �

5. Polynomial growth and bounded packing

In this section we observe the basic fact that Borel actions of finitely-generatedgroups of polynomial volume-growth have finite Borel asymptotic dimension. Wealso explore notions of bounded packing, some of which will be significant to thenext section.

Our work in the first half of this section, dealing with groups of polynomialvolume-growth, essentially replicates the work of Jackson–Kechris–Louveau [19]but is written in the the slightly different language of asymptotic dimension.

Definition 5.1. We say that a proper extended metric space (X, ρ) has boundedpacking of degree d if for every t > 1 and r0 > 0 there is r ≥ r0 so that for allx ∈ X, B(x; rt) contains at most 2(t+ 1)d-many pairwise disjoint balls of radius r.

Lemma 5.2. Let G be a finitely generated group with polynomial volume-growth ofdegree d, and let B be a finite symmetric generating set for G with 1G ∈ B. Thenfor every m ≥ 1 and r0 > 0 there is r ≥ r0 so that |Brm| ≤ 2md|Br|.

Proof. Fix m ≥ 1, r0 > 0, and let c > 0 satisfy |Br| ≤ crd for all r ≥ 1. Towards acontradiction, suppose |Brm| > 2md|Br| for every r ≥ r0. Then by induction, forevery k ∈ N

cmkdrd0 ≥ |Bmkr0 | > 2kmkd|Br0 | ≥ 2kmkd,

which is a contradiction when crd0 < 2k. �

Recall from Section 2 that if B is any finite generating set for G then we obtaina proper right-invariant metric τB on G. Furthermore, for an action G y X weobtain a proper extended metric ρτB on X.

Corollary 5.3. Let G be a finitely-generated group with polynomial volume-growthof degree d, let B be a finite symmetric generating set for G with 1G ∈ B, and letGy X be any action. Then (X, ρτB ) has bounded packing of degree d.

Proof. Fix t > 1 and r0 > 0. Set m = t + 1 and let r ≥ r0 be as given by Lemma5.2. Fix x ∈ X and suppose that the balls B(yi; r), i ∈ k, are pairwise disjointand contained in B(x; rt) = Brt · x. Pick gi ∈ Brt with gi · x = yi. Since the setsB(yi; r) = Brgi · x are pairwise disjoint, the sets Brgi must be pairwise disjoint

subsets of Br(t+1) = Brm. Therefore k ≤ |Brm||Br| ≤ 2md = 2(t+ 1)d. �


Lemma 5.4. Let (X, ρ) be a proper Borel extended metric space with boundedpacking of degree d. Then asdimB(X, ρ) < 2 · 5d <∞.

Proof. Since ρ is proper, Eρ is countable so there are Borel functions fn : X → Xwith Eρ = {(x, fn(x)) : x ∈ X, n ∈ N}. Fix r0 > 0, set t = 4, and let r ≥ r0 be asin the definition of bounded packing. We will verify condition (1) of Lemma 3.1.Consider the Borel graph Γ where x, y ∈ X are joined by an edge if 0 < ρ(x, y) ≤ 2r.Then Γ is locally finite since ρ is proper, so by [21, Prop. 4.2 and Prop. 4.3] thereis a Γ-independent Borel set Y ⊆ X that is maximal, meaning every x ∈ X \ Y isadjacent to some y ∈ Y . Therefore the sets B(y; r), y ∈ Y , are pairwise disjointand B(Y ; 2r) = X. Now define s : X → Y by setting s(x) = fk(x) if there isk ∈ N with fk(x) ∈ Y and ρ(x, fk(x)) ≤ r, and otherwise setting s(x) = fk(x)where k ∈ N is least with fk(x) ∈ Y and ρ(x, fk(x)) ≤ 2r. Lastly, define E byx E x′ ⇔ s(x) = s(x′). Then E is Borel and every class has diameter at most 4r,so E is uniformly bounded. Also, for any x ∈ X the sets B(y; r), y ∈ s(B(x; r)), arepairwise disjoint and contained in B(x; 4r), so the number of E-classes that meetB(x; r) is |s(B(x; r))| ≤ 2 · 5d. �

Corollary 5.5. If G is a countable group having uniform local polynomial volume-growth of degree d, X is a standard Borel space, and G y X is any Borel action,then asdimB(Gy X) < 2 · 5d <∞.

Proof. This follows from Corollary 2.2 (which extends to Borel asymptotic dimen-sion, as noted in the final paragraph of Section 3), Corollary 5.3, and Lemma5.4. �

We next explore a variant notion of bounded packing that applies to groupextensions and will help us to move beyond groups of polynomial volume-growth.The form of the next definition (the use of B3 in particular) is tailored to thespecific needs we will encounter in the next section.

Definition 5.6. Let G be a countable group and let Φ be a finite set of automor-phisms of G. We say that G has Φ-bounded packing if there is k with the followingproperty: for every finite set B0 ⊆ G there is a finite symmetric set B ⊇ B0 ∪{1G}such that |T | ≤ k whenever φ ∈ Φ, T ⊆ B3φ(B3) and (TT−1 \ {1G}) ∩B = ∅.

Lemma 5.7. Let G be a countable group and Φ a finite set of automorphisms ofG. Assume that either:

(i) G has uniform local polynomial volume-growth and Φ = {id} consists of thetrivial automorphism;

(ii) G is a finite group or a countable torsion-free abelian group with finite Q-rank; or

(iii) G =⊕

n∈Z Z2 and each φ ∈ Φ shifts the Z-coordinates of G by some s ∈ Z.

Then G has Φ-bounded packing.

Proof. (i). Say G has uniform local polynomial volume-growth of degree at mostd and set k = 2 · 12d. Let B0 ⊆ G be finite. Without loss of generality we canassume B0 is symmetric and contains 1G. Apply Lemma 5.2 to obtain r such that|B12r

0 | ≤ 2(12)d|Br0 |, and set B = B2r0 . If T ⊆ B6 satisfies (TT−1 \ {1G}) ∩B = ∅

then the sets Br0 · t, t ∈ T , are pairwise disjoint subsets of B12r0 and hence |T | ≤

|B12r0 |/|B0| ≤ k.


(ii). When G is finite this is trivial; simply take B = G and k = 1. So considerthe other case and let d be the Q-rank of G. Let A be a finite symmetric set thatcontains 1G and generates a subgroup G0 of Q-rank d. Then every element of G/G0

has finite order, and since G1 = 〈G0 ∪⋃φ∈Φ φ(G0)〉 is finitely generated, G1/G0

must be finite. Pick a finite set ∆ containing 1G with G0∆ = G1, and pick λ ∈ Nwith (

⋃φ∈Φ ∆φ(A)∆−1) ∩G0 ⊆ Aλ. Then for φ ∈ Φ, r ∈ N, and a1, . . . , ar ∈ A, if

we pick δi ∈ ∆ with φ(a1a2 · · · ai) ∈ G0δi we obtain

φ(a1 · · · ar) = (φ(a1)δ−11 )(δ1φ(a2)δ−1

2 ) · · · (δr−1φ(ar)δ−1r )δr ∈ Aλr∆.

Therefore φ(Ar) ⊆ Arλ∆ for all φ ∈ Φ and r ∈ N.Set k = 2(7 + 6λ)d|∆| and let B0 ⊆ G be finite. Notice that since G is a torsion-

free abelian group, for every m ≥ 1 the map πm : G → G given by πm(g) = gm isan injective homomorphism. Since every element of G/G0 has finite order, there ism ≥ 1 with πm(B0) ⊆ G0. As finitely generated abelian groups of Q-rank d havepolynomial volume-growth of degree at most d, we can apply Lemma 5.2 to obtainr ∈ N with Ar ⊇ πm(B0) and |A(7+6λ)r| ≤ 2(7 + 6λ)d|Ar|. Now set B = π−1

m (A2r).Then B is finite, symmetric, and contains B0 ∪ {1G}. Lastly, consider φ ∈ Φ andsuppose T ⊆ B3φ(B3) satisfies (TT−1 \ {1G}) ∩B = ∅. Then the sets Ar · πm(t),t ∈ T , are pairwise disjoint subsets of A7rφ(A6r) ⊆ A(7+6λ)r∆. Therefore

|T | ≤ |A(7+6λ)r||Ar|

· |∆| ≤ 2(7 + 6λ)d|∆| = k.

(iii). Suppose that every φ ∈ Φ shifts the Z-coordinates by at most s ∈ Nunits. Set k = 22s and let B0 ⊆ G be finite. For m ∈ N define the subgroupFm =

⊕mn=−m Z2. Pick any m ≥ s with B0 ⊆ Fm and set B = Fm. Then B3 = B

and φ(B) ⊆ Fm+s for every φ ∈ Φ. Let φ ∈ Φ and consider T ⊆ B3φ(B3) satisfying(TT−1 \ {1G}) ∩B = ∅. Then the sets Fmt, t ∈ T , are pairwise disjoint subsets ofFm+s and hence |T | ≤ |Fm+s : Fm| = 22s = k. �

The notion of Φ-bounded packing immediately leads to a packing property forgroup extensions.

Lemma 5.8. Let G � H be countable groups and let C ⊆ H be finite. Assumethat G has {φc : c ∈ C}-bounded packing where φc(g) = cgc−1. Then there is kGwith the following property: for any finite set B0 ⊆ G there is a finite symmetricset B0 ⊆ B ⊆ G with 1H ∈ B such that |T | ≤ k whenever T ⊆ B3CB3 satisfies(TT−1 \ {1G}) ∩B = ∅.

Proof. Set kG = k|C| where k is as in Definition 5.6. Let B0 ⊆ G be finite.By assumption there is a finite symmetric set B0 ⊆ B ⊆ G with 1G ∈ B andwith the property that |S| ≤ k whenever c ∈ C and S ⊆ B3cB3c−1 satisfies(SS−1 \ {1G}) ∩ B = ∅. Now let T ⊆ B3CB3 satisfy (TT−1 \ {1G}) ∩ B =∅. Then for c ∈ C the set Tc = (T ∩ B3cB3)c−1 satisfies Tc ⊆ B3cB3c−1 and(TcT

−1c \ {1G}) ∩B = ∅. Therefore |Tc| ≤ k and |T | ≤ k|C| = kG. �

For our work with normal series in the next section, we will need the followingrelativized version of the previous lemma.

Corollary 5.9. Let H be a countable group, let F � G be normal subgroups ofH, and let C ⊆ H be finite. Assume that G/F has {φc : c ∈ C}-bounded packingwhere φc(gF ) = cgc−1F , and let kG/F be as in Lemma 5.8. Then for any finite set


B0 ⊆ G there are finite symmetric sets A ⊆ F and B0 ⊆ B ⊆ G with 1H ∈ A ∩ Bsuch that |T | ≤ kG/F whenever T ⊆ B3CB3 satisfies (TT−1 \ {1G}) ∩ABA = ∅.

Proof. It suffices to observe that for any finite set B ⊆ G containing 1H thereis a finite symmetric set A ⊆ F containing 1H such that whenever T ⊆ B3CB3

and (TT−1 \ {1H}) ∩ ABA = ∅, we have that T maps injectively into H/F and(TT−1 \ {1H}) ∩ BF = ∅. Indeed, as long as A contains F ∩ B4CB6C−1B3, wewill have that when t1, t2 ∈ B3CB3 satisfy t1t

−12 ∈ BF then

t1t−12 ∈ BF ∩ (B3CB6C−1B3) ⊆ B(F ∩B4CB6C−1B3) ⊆ BA ⊆ ABA. �

6. Normal series

Our goal in this section is to prove that if a group G has a suitable normal series{1G} = Gn�Gn−1 � · · ·�G0 = G then all free Borel actions of G have finite Borelasymptotic dimension. Our argument will be inductive and will ascend throughthe normal series. However, even in the simplest situation of an extension by Z, wedo not know of any property of G-actions that both is preserved by Z-extensionsand implies all free Borel G-actions have finite Borel asymptotic dimension. Ourinductive assumption will therefore need to be forward-looking and will be a state-ment about both the kth term Gk and the whole group G. The condition we needis specified in the definition below.

For an action G y X and B ⊆ G, we say that Y ⊆ X is B-independent ifB · Y ∩ Y = ∅

Definition 6.1. Let G � H be countable groups. We say the pair (G,H) hasgood asymptotics if for every standard Borel space X and every free Borel actionH y X, the restricted action G y X has finite Borel asymptotic dimension andfor every finite set C ⊆ H there is ` ∈ N so that for every finite set B ⊆ G there isa Borel cover {W0, . . . ,W`} of X consisting of CB \G-independent sets.

Ultimately we will show that when F�G are normal subgroups ofH, if (F,H) hasgood asymptotics and G/F has a suitable form then (G,H) has good asymptotics.This will require us to structure H-actions in a way that diminishes the dynamicsof F and emphasizes the structure of H/F . The following lemma achieves this, aswe will explain before going into the proof.

Lemma 6.2. Let F �H be countable groups with (F,H) having good asymptotics.Let X be a standard Borel space, let H y X be a free Borel action, and set d =asdimB(F y X) <∞. Then for any finite sets C ⊆ H and A ⊆ F there is a Borelcover {Uni : i ∈ d + 1, n ∈ ` + 1} of X such that Uni is CA \ F -independent andFA(Uni ) is uniformly finite, and there are finite sets CA ∩ F ⊆ An ⊆ F such thatwhenever Z is a FAn

(Uni )-class, c ∈ C, and m > n, the set cA ·Z is either disjointwith Umi or else contained in a single FAm

(Umi )-class.

The most important feature of this lemma is the final property, which is bestillustrated in the case that 1H ∈ A and C−1C∩F ⊆ A. Specifically, since C∩cF ⊆cA, for any FAn

(Uni )-class Z and any FAm(Umi )-class Z ′ with m > n, whether c·Z is

contained in or disjoint with Z ′ only depends on the coset cF ∈ CF/F . Moreover,C ·Z is disjoint with Uni \Z since Uni is C \F -independent and C∩F ⊆ An. Thus ina certain local (only considering transformations by the elements of C) and ordered(requiring n ≤ m) sense, we are able to view X in a way that allows dynamicaldata to descend to the quotient H/F .


The basic idea of the proof is that we will take a Borel cover {W0, . . . ,W`} ofX consisting of (C−1CA′) \ F -independent sets for some carefully chosen A′ ⊆ F .Using the fact that asdimB(F y X) = d, we will choose a Borel cover {V n0 , . . . , V nd }of Wn for each n ∈ ` + 1. Moreover, we will arrange for FQn

(V ni ) to be uniformly

finite, where Qn ⊇ An is chosen to be large relative to the shapes of the FQk(V kj )-

classes for k < n, j ∈ d + 1. In other words, the Qn’s signify a rapid growth inscales. We will then slightly enlarge each set V ni ∩Wn to Uni ⊆ An · (V ni ∩Wn) ina process quite similar to the proofs of Lemma 4.5 and Corollary 4.6. Specifically,whenever D is a FQn

(V ni ∩Wn)-class, c ∈ C, and A−1n (C∩cF )An ·D comes close to

V mi ∩Wm withm > n, we will arrange that Umi contain cAn·D. For this enlargementprocedure to be stable, we need the sets A−1

n (C ∩ cF )An ·D to be pairwise disjoint.Such disjointness will be derived from the independence properties of the Wn’s.

Proof. Without loss of generality, we may assume 1H ∈ A ∩ C. Let ` be as inDefinition 6.1 for the set C−1C. Set A0 = A ∪ (CA ∩ F ). Inductively assume thatAn ⊆ F has been defined. Set Tn = A−1

n CAn and Qn = A−1n A2

n∪ (T−1n Tn∩F ). Let

{V n0 , . . . , V nd } be a Borel cover ofX such that FQn(V ni ) is uniformly finite for every i.

Now pick a finite set An+1 ⊆ F containing An and satisfying AnA−1n (C∩cG)An·Y ⊆

An+1 · x for every FQn(V ni )-class Y , c ∈ C, and x ∈ A−1

n cAn · Y .

Since T−1n Tn ⊆ C−1CF for every n, by our choice of ` there is a Borel cover

{W0, . . . ,W`} of X so that Wn is T−1n Tn \ F -independent for every n. We will

construct Uni ⊇ V ni ∩Wn. As preparation, for X ′ ⊆ X let sni (X ′) be the smallestset containing X ′ and containing A−1

n (C ∩ cF )An · Y whenever c ∈ C and Y is aFQn

(V ni ∩Wn)-class satisfying A−1n cAn ·Y ∩X ′ 6= ∅. Notice that if X ′ is Borel then

so is sni (X ′), as x ∈ sni (X ′) if and only if there is c ∈ C, t1, t2 ∈ A−1n (C ∩ cF )An,

and a ∈ An+1 with a · x ∈ X ′, t−11 · x, t−1

2 a · x ∈ V ni ∩ Wn, and (t−11 · x, t−1

2 a ·x) ∈ FQn

(V ni ∩Wn). Also notice that since Wn is T−1n Tn \ F -independent the sets

A−1n (C ∩ cF )An · (V ni ∩Wn) are pairwise disjoint for distinct cosets cF ∈ CF/F ,

and since additionally T−1n Tn ∩ F ⊆ Qn, we have that Tn · Y ∩ Tn · Y ′ = ∅ for all

distinct classes Y 6= Y ′ of FQn(V ni ∩Wn). Therefore sni ◦ sni (X ′) = sni (X ′).

Define the Borel sets U0i = V 0

i ∩ W0 and Uni = s0i ◦ · · · ◦ s

n−1i (V ni ∩ Wn) for

n ≥ 1. Then V ni ∩Wn ⊆ Uni so the sets Uni cover X. Also, by definition of An+1 wehave An · sni (X ′) ⊆ An+1 ·X ′, and it follows from induction and the containment1H ∈ A ⊆ A0 that Uni ⊆ A · Uni ⊆ An · (V ni ∩Wn). Thus each set Uni is (CA) \A-independent since AUni ⊆ An ·Wn, and each relation FA(Uni ) is uniformly finitesince Uni ⊆ An · V ni and FQn

(V ni ) ⊇ FA−1

n AAn(V ni ) is uniformly finite.

Lastly, consider a FAn(Uni )-class Z. Set Y0 = (A−1

n A · Z) ∩ V ni ∩ Wn. Then

Z ⊆ A · Z ⊆ An · Y0 and, since A−1n A2

n ⊆ Qn, Y0 must be contained in a singleFQn

(V ni ∩Wn)-class Y . So A−1n cA · Z ⊆ A−1

n (C ∩ cF )An · Y . For m > n the set

sni ◦ · · · sm−1i (V mi ∩Wm) either contains A−1

n (C ∩ cF )An · Y or is disjoint with it.

As Umi ⊆ An ·sni ◦ · · · ◦sm−1i (V mi ∩Wm), it follows that Umi either contains cA ·Z or

is disjoint with it. Finally, when cA ·Z is a subset of Umi it is contained in a singleFAm

(Umi )-class since cA · Z ⊆ A−1n cAn · Y ⊆ Am · x for all x ∈ A−1

n cAn · Y . �

The next lemma provides the crucial inductive step that will allow us to ascendthrough a normal series.

Lemma 6.3. Let H be a countable group and let F �G be normal subgroups of H.Assume that (F,H) has good asymptotics and that G/F has {φc : c ∈ C}-bounded


packing whenever C ⊆ H is a finite set with the property that each automorphismφc(gF ) = cgc−1F is either trivial or an outer automorphism. Then (G,H) hasgood asymptotics.

Proof. Let X be a standard Borel space, let H y X be a free Borel action, let C ⊆H be a finite symmetric set containing 1H , and set dF = asdimB(F y X) < ∞.From Definition 6.1 we see that without loss of generality we can replace each c ∈ Cwith any element of cG. We can therefore assume that the conjugation of each c ∈ Con G either is an outer automorphism or is trivial, while still maintaining that Cis symmetric and contains 1H . Our assumptions then allow us to apply Corollary5.9 to obtain kG/F ∈ N. Set `G = (kG/F + 1)(dF + 1), fix a finite set B0 ⊆ G, andlet A ⊆ F and B0 ⊆ B ⊆ G be finite symmetric sets containing 1H as described inCorollary 5.9.

Apply Lemma 6.2 to obtain finite sets (AB3A)∩F ⊆ An ⊆ F and a Borel cover{Uni : i ∈ dF + 1, n ∈ `F + 1} of X such that each set Uni is (ABA ∪B3CB3) \ F -independent, each relation FB3∩F (Uni ) is uniformly finite, and whenever Z is aFAn

(Uni )-class, t ∈ B3CB3, and m > n, the set t((BA)2 ∩ F ) · Z is either disjointwith Umi or else contained in a single FAm

(Umi )-class. Next define Borel sets V ni ⊆Uni as follows: set V `Fi = U `Fi and by reverse induction on n ∈ `F + 1 define

V ni = Uni \⋃m>nB · V mi .

Notice that since 1H ∈ B the collection {B · V ni : i ∈ dF + 1, n ∈ `F + 1} coverseach Uni and thus covers X. Additionally, it is simple to see by reverse inductionon n that V ni is a FAn

(Uni )-invariant subset of Uni . Consequently, the followingproperties hold:

(i) if Z is a FAn(V ni )-class then for every m > n and t ∈ B3CB3 we have that

t · Z is either disjoint with V mi or else contained in a FAm(V mi )-class;

(ii) if Z and Z ′ are classes of FAn(Uni ) and FAm

(Umi ) respectively and Z 6= Z ′,then ABA · Z ∩ Z ′ = ∅.

Statement (i) holds since each V ni is FAn(Uni )-invariant, and when n = m statement

(ii) holds since Uni ⊇ V ni is (ABA) \ F -independent and (ABA) ∩ F ⊆ An. Nowsuppose Z and Z ′ are as described in (ii) with n 6= m. Since ABA = (ABA)−1, wecan assume m > n. Then for every b ∈ B we have that AbA ·Z = b(b−1Ab)A ·Z ⊆b((BA)2 ∩ F ) · Z is either disjoint with or contained in Z ′, but the containmentb · Z ⊆ AbA · Z ⊆ Z ′ is prohibited by definition of V ni .

Define a directed Borel graph Γi on⊔n∈`F +1 V

ni by placing an edge directed

from x to y if x ∈ B3CB3 · y (equivalently y ∈ B3CB3 · x) and there are n < mwith x ∈ V ni and y ∈ V mi . We construct a proper Borel coloring πi :

⊔n∈`F +1 V

ni →

kG/F +1 of Γi as follows. First let πi have constant value 0 on V `Fi . Now inductivelyassume that πi is defined on V mi and is FAm

(V mi )-invariant for every m > n. Given

any x ∈ V ni , let T ⊆ B3CB3 be such that for each m > n and FAm(V mi )-class Z

receiving an edge from x, there is exactly one t ∈ T with t · x ∈ Z. Then (ii) andCorollary 5.9 imply that |T | ≤ kG/F , and our inductive hypothesis implies that thenumber of πi-values used by forward-neighbors of x is at most |T |. Moreover, (i)and our inductive hypothesis imply that for all (x, y) ∈ FAn

(V ni ) and t ∈ B3CB3,either πi(t ·x) = πi(t ·y) or else neither is defined. Thus if for each x ∈ V ni we defineπi(x) be the least element of kG/F + 1 not used by any of the forward-neighbors ofx, then πi � V ni is Borel and FAn

(V ni )-invariant.


To finish the proof, set W ji = B · π−1

i (j). Then we have `G-many Borel sets W ji

that cover X. Notice that every w ∈W ji is a B-translate of some v ∈

⊔n∈`F +1 V

ni

with πi(v) = j. Therefore, since πi is a proper coloring, whenever w,w′ ∈ W ji

satisfy w ∈ B2CB2 · w′, there must be a single n with w,w′ ∈ B · V ni . It follows

that FB(W ji ) is uniformly finite since each set Uni ⊇ V ni is B3 \F -independent and

each relation FB3∩F (Uni ) ⊇ FB3∩F (V ni ) is uniformly finite. As `G was specified

before B, we find asdimB(Gy X) ≤ `G − 1 <∞. Finally, each set W ji is CB \G-

independent since each set Uni ⊇ V ni is (BCB2)\F -independent. We conclude that(G,H) has good asymptotics. �

Theorem 6.4. Let G be a countable group admitting a normal series where eachquotient of consecutive terms is a finite group or a torsion-free abelian group withfinite Q-rank, except the top quotient can be any group of uniform local polynomialvolume-growth or the lamplighter group Z2 o Z. Then for every free Borel actionGy X on a standard Borel space X we have asdimB(Gy X) = asdim(G) <∞.

We remark that the quotients of consecutive terms can be allowed to have a moregeneral form. Specifically, by refining the normal series it is enough to assumethat each quotient of consecutive terms, aside from the top quotient, admits acharacteristic series where each consecutive quotient is finite or torsion-free abelianwith finite Q-rank. For instance, abelian groups that have a finite torsion subgroupand finite Q-rank admit such a characteristic series, as do all finitely generatednilpotent groups and all polycyclic groups, so one can allow these groups to appearas consecutive quotients.

Proof. Let the normal series be {1G} = Gn � Gn−1 � · · · � G0 = G. In the casewhere the top quotient is the lamplighter group, we may enlarge the normal seriesby one term and re-index so that G0/G2 is isomorphic to Z2 oZ via an isomorphismidentifying G1/G2 with

⊕n∈Z Z2. Since asdim(G y X) = asdim(G) for for all

free actions, by Theorem 4.2 we only need to show asdimB(G y X) < ∞ forall free Borel actions. So it suffices to prove that (Gk, G) has good asymptoticsfor k = n, n − 1, . . . , 0. For (Gn, G) = ({1G}, G) this is trivial, since for any freeBorel action G y X we clearly have asdimB({1G} y X) = 0 and for any finiteC ⊆ G we can apply [21, Prop. 4.6] to partition X into (|C| + 1)-many Borelsets that are C \ {1G}-independent. For the inductive step, suppose (Gk+1, G) hasgood asymptotics for some n > k ≥ 0. If Gk/Gk+1 is finite or is a torsion-freeabelian group with finite Q-rank, then Lemmas 5.7.(ii) and 6.3 imply that (Gk, G)has good asymptotics (note this case applies when G0/G2 is the lamplighter groupand k = 0). If k = 1 and G0/G2 is the lamplighter group, then every outerautomorphism of G1/G2 induced by G is of the form described in Lemma 5.7.(iii)and thus that lemma and Lemma 6.3 imply that (G1, G) has good asymptotics.Finally, if k = 0 and G0/G1 has uniform local polynomial volume-growth then Ginduces no outer automorphisms of G0/G1 and thus Lemmas 5.7.(i) and 6.3 implythat (G0, G) has good asymptotics. �

Clearly the above theorem applies to all polycyclic groups and the lamplightergroup Z2 oZ. We remark that it applies to the Baumslag–Solitar group BS(1, 2) aswell since it is an extension of the group of dyadic rationals (a torsion-free abeliangroup of Q-rank 1) by Z.


If in Theorem 6.4 we additionally allowed consecutive quotients to be locallyfinite, then to the best of our knowledge this theorem would cover all countableamenable groups currently known to have finite standard asymptotic dimension.Since Lemma 5.7 fails for many locally finite groups, a different method would beneeded to prove this stronger version. On the other hand, Theorem 6.4 cannot beextended to any non-amenable group as this would contradict Theorem 7.1 sincesuch groups admit free actions whose orbit equivalence relation is not hyperfinite[19].

Another way of improving the above theorem would be to extend it to hold forall (not necessarily free) Borel actions. Hypothetically this might be possible whilekeeping most of our methods intact, but we were not able to find any way of doingthis.

7. Hyperfinite equivalence relations

Our initial motivation for this present work was to study hyperfinite equivalencerelations with the goal of providing a positive answer to Weiss’ question in the caseof polycyclic groups. It was through working towards that goal that the notionof Borel asymptotic dimension slowly but naturally emerged. Perhaps one of themost surprising aspects of this work is the discovery of just how well-suited Borelasymptotic dimension is to the study of hyperfinite equivalence relations, and howin hindsight it may have played an important but hidden role in past investigationsinto Weiss’ question.

We first present the simple argument that Eρ is hyperfinite whenever ρ is properand (X, ρ) has finite Borel asymptotic dimension.

Theorem 7.1. Suppose that (X, ρ) is a proper Borel extended metric space, andset Eρ = {(x, x′) ∈ X × X : ρ(x, x′) < ∞}. If (X, ρ) has finite Borel asymptoticdimension then Eρ is hyperfinite.

Proof. We will rely on condition (1) of Lemma 3.1. Let d < ∞ be the Borelasymptotic dimension of (X, ρ). We will inductively build radii rn ≥ n and anincreasing sequence of uniformly bounded Borel equivalence relations Fn with theproperty that for every n and x ∈ X, B(x; rn) meets at most d + 1 classes of Fn.To begin set r0 = 0 and let F0 be the equality relation on X. Inductively assumethat Fn−1 has been constructed. Then Fn−1 is finite since ρ is proper. So thereis a Borel selector s : X → X satisfying s(y) = s(x) Fn−1 x for all (x, y) ∈ Fn−1.Choose rn ≥ n so that [x]Fn−1

⊆ B(x; rn) for all x ∈ X, and choose a uniformlybounded Borel equivalence relation E with the property that for every x, B(x; 2rn)meets at most d+ 1 classes of E. Now define Fn = {(x, y) ∈ X ×X : s(x) E s(y)}.Then Fn is Borel, uniformly bounded, and contains Fn−1. Additionally, for any x,the number of Fn classes meeting B(x; rn) is bounded by the number of E-classesmeeting B(x; 2rn) ⊇ s(B(x; rn)), which is at most d + 1. Now set F∞ =

⋃n Fn.

Then F∞ is hyperfinite. For every n and x, B(x;n) meets at most d+ 1 classes ofFn and hence at most d+1 classes of F∞. Thus every Eρ-class divides into at mostd+ 1 classes of F∞, which implies Eρ is hyperfinite as well [19, Prop. 1.3.(vii)]. �

The above proof together with our proof of Corollary 5.5 is essentially identicalto the argument of Jackson–Kechris–Louveau that Borel actions of groups of poly-nomial volume-growth generate hyperfinite equivalence relations [19]. It is naturalto wonder then why the utility of Borel asymptotic dimension was not identified


earlier. We suspect that part of the reason is that the above argument relies oncondition (1) of Lemma 3.1 while the discoveries we make here depend heavily oncondition (2) of that lemma, and the implication (1) ⇒ (2) is not immediatelyobvious.

When we consider the overlap between Weiss’ question and the Union Prob-lem, the significance of Borel asymptotic dimension becomes much more apparent.Special cases of this overlap were solved in the works of Gao–Jackson [11] andSchneider–Seward [27] through highly technical arguments that spanned dozens ofpages. However, in the next theorem we will see that a much more general resultcan be obtained using Borel asymptotic dimension and that the proof is less tech-nical and tremendously shorter in comparison. Furthermore, we believe that ourproof here preserves the core concept of those prior works, as we now discuss.

The key concept in [11] and [27] is the notion of orthogonality. In those works,orthogonality is a property for sequences of finite Borel equivalence relations Enthat are contained in a fixed orbit equivalence relation EXG , and the property func-tions to enforce that the boundaries of the En’s within EXG vanish as n → ∞. Inthis situation, EXG will be seen to be hyperfinite as it will be equal to the unionof the increasing sequence of equivalence relations

⋂k≥nEk. Roughly speaking,

orthogonality was defined by using the geometry of the group G to define varioustypes of facial boundaries and then requiring that every facial boundary of En befar (in a manner determined by n) from the corresponding facial boundary of Emwhen m > n. For instance, when G = Zd and e1, . . . , ed are the standard gen-erators of Zd, the type i boundary of En is the set of x with (x, ei · x) 6∈ En or(x,−ei · x) 6∈ En.

We believe that the essence of orthogonality can be captured in the languageof asymptotic dimension. Instead of seeking separation of various facial boundarytypes in order to guarantee the vanishing of boundaries in the limit, one can considersequences of (d+1)-indexed covers and require that for each index i the boundariesof the corresponding sets are separated from each other. In this sense, we alreadyconstructed orthogonal sequences in Corollary 4.6, and this construction was far lesstechnical than those in [11, 27]. Before proceeding, we record a simplified versionof that corollary.

Corollary 7.2. Let (X, ρ) be a Borel extended metric space with Eρ countable andwith Borel asymptotic dimension d <∞. Then for any r > 0 and D ∈ N there existBorel covers Vj = {V j0 , . . . , V

jd } of X for j ∈ D + 1 such that Fr(V

ji ) is uniformly

bounded for all i and j, and for every x ∈ X and i ∈ d + 1 there is at most onej ∈ D + 1 such that V ji divides B(x; r).

Proof. By using rj = 2r, such a sequence of covers is immediately obtained fromCorollary 4.6. �

In [11, 27] the notion of orthogonality (using facial boundary types as discussedabove) was defined for each of the finitely generated subgroups of G and, in con-fronting the Union Problem, the ultimate technical challenge was to combine thesenotions together by diagonalizing over all the finitely generated subgroups of G.This involved an intricate and cumbersome inductive argument that was chore-ographed through a 2-dimensional triangular array. In contrast, with Borel asymp-totic dimension we are able to perform this diagonalization in a different way that


is much more automated and direct. We put all of these pieces together in thetheorem below.

Theorem 7.3. Let X be a standard Borel space and let (ρn)n∈N be a sequenceof proper Borel extended metrics on X. Assume for every n ∈ N and r > 0 thatsup{ρn+1(x, y) : x, y ∈ X ρn(x, y) < r} is finite and set E =

⋃nEρn . If (X, ρn)

has finite Borel asymptotic dimension for every n ∈ N then E is hyperfinite.

Proof. Inductively for n ∈ N, pick rn ≥ n large enough that Bρk(x;n + rk) ⊆Bρn(x; rn) for all k < n and x ∈ X. So we have (x, y) ∈ E ⇔ ∃n ρn(x, y) ≤ rn.Also let dn be the Borel asymptotic dimension of (X, ρn). Inductively we will buildfinite Borel covers (Un)n∈N, (Vn,j)n∈N,j∈dn+1+1 of X satisfying the following:

(i) Fρn2rn(U) is ρn-uniformly bounded for every U ∈ Un ∪

⋃j∈dn+1+1 Vn,j ;

(ii) for every k ∈ N, n ≥ k, and x ∈ X there is at most (dk + 1)-many coversW ∈ {Um : k ≤ m ≤ n} ∪ {Vjn : j ∈ dn+1 + 1} that divide Bρk(x; rk)(meaning there is W ∈ W that divides Bρk(x; rk)).

The role of the Vn,j ’s will be transient and only used to inductively build the Un’s.Denote by Fn the Borel equivalence relation where (x, x′) ∈ Fn if and only if forevery U ∈ Un we have either x, x′ 6∈ U or (x, x′) ∈ Fρn2rn

(U). Then (i) implies thatFn is finite. If (x, y) ∈ E then by construction there is k with ρk(x, y) ≤ rk, and sofrom (ii) we see that (x, y) ∈ Fn for all but at most (dk+1)-many n ≥ k. Thus, oncethese covers are constructed we will have that E =

⋃k

⋂n≥k Fn is hyperfinite. To

begin the construction, apply Corollary 7.2 to obtain Borel covers U0, (V0,j)j∈d1+1

satisfying (i) and (ii) with k = n = 0.Now suppose that U0, . . . ,Un−1, Vn−1,0, . . . ,Vn−1,dn have been constructed. By

enlarging rn if necessary, we can assume that the ρn-diameter of the classes ofFρn−1

2rn−1(V ) are bounded by rn for every j and V ∈ Vn−1,j . Apply Corollary 7.2 to

obtain a sequence of Borel covers Vn,j = {V n,j0 , . . . , V n,jdn} for j ∈ dn+1 + 2 with the

property that Fρn4rn(V n,ji ) is ρn-uniformly bounded for all i and j, and such that for

every x ∈ X and i ∈ dn + 1 there is at most one j ∈ dn+1 + 2 with V n,ji dividingB(x; 2rn).

Now for j ∈ dn+1 + 2 define Vn,j = {V n,j0 , . . . , V n,jdn} where V n,ji is the Borel

set obtained by taking the union, over all V ∈ Vn−1,i, of the Fρn−1

2rn−1(V )-classes

meeting V n,ji . Since Vn−1,i covers X we have V n,ji ⊇ V n,ji and thus Vn,j covers X.

Also, since V n,ji ⊆ Bρn(V n,ji ; rn) it is immediate that Fρn2rn(V n,ji ) is ρn-uniformly

bounded.Define Un = Vn,dn+1+1. It only remains to check (b). So fix k ≤ n and x ∈ X. If

k = n then this is immediate since Vn,j divides Bρn(x; rn) only when Vn,j dividesBρn(x; 2rn). So assume k < n. Let J be the set of j ∈ dn+1 + 2 such that Vn,jdivides Bρk(x; rk) and let I be the set of i ∈ dn + 1 such that Vn−1,i dividesBρk(x; rk). By our inductive assumption it suffices to show that |J | ≤ |I|.

Fix j ∈ J and let t(j) = i be any element of dn + 1 such that V n,ji divides

Bρk(x; rk). By definition of V n,ji , there is some V ∈ V n−1,i with a class of Fρn−1

2rn−1(V )

dividing Bρk(x; rk). This is only possible if V divides Bρk(x; rk) as the latter set

has ρn−1-diameter at most 2rn−1. Thus i ∈ I. Additionally, V n,ji must divideBρn(x; 2rn), and with i and x fixed there is at most one such value of j ∈ dn+1 + 2with this property. Thus t is an injection from J to I. �


Corollary 7.4. Suppose that G is the union of groups satisfying the assumption ofTheorem 6.4. Then for every free Borel action G y X on a standard Borel spaceX, the orbit equivalence relation EXG is hyperfinite.

The above corollary applies to all locally polycyclic groups, the Baumslag–Solitargroup BS(1, 2), the lamplighter group Z2 o Z, and many groups not having finitestandard asymptotic dimension, such as (

⊕n∈N Z) o S∞ where S∞ is the group of

finitely supported permutations of N. Moreover, the restriction to free actions canbe removed in the case of polycyclic groups by [27, Cor. 8.2] and in the case of localpolynomial volume-growth by Corollary 5.5. We thus obtain a positive solution toWeiss’ question in the case of polycyclic groups and groups of local polynomialvolume-growth.

Corollary 7.5. Let G be either a polycyclic group or a countable group of localpolynomial volume-growth. If X is a standard Borel space and Gy X is any Borelaction, then the orbit equivalence relation EXG is hyperfinite.

These two corollaries could be strengthened if Theorem 6.4 were strengthened (aswe discussed at the end of Section 6). However, further progress on Weiss’ questionwill eventually involve groups that do not have finite standard asymptotic dimensionnor are unions of such groups. For instance, Z o Z and the Grigorchuk group arenot unions of groups having finite standard asymptotic dimension. Looking at thebig picture, we feel that Borel asymptotic dimension is more than a mere propertythat implies hyperfiniteness, but rather the language surrounding this concept ismeaningful and efficient at handling certain tasks related to hyperfiniteness proofs,as displayed in the proof of Theorem 7.3. Thus, while the ability of Borel asymptoticdimension is strictly limited in providing further advances on Weiss’ question, wesee potential for the language surrounding this notion to evolve to handle moregeneral situations.

8. Borel graphs and chromatic numbers

The objects as in Lemma 3.1.(2’) that witness asi(X, ρ) ≤ 1 are somewhat similarto objects called “barriers” in [6] and “toast” in [12, 13, 14]. In [6, 14] those objectswere used to obtain bounds on Borel chromatic numbers. Below we generalize theproof by Conley–Miller in [6] to our setting of asymptotic separation index. Apeculiar aspect to our generalization is that we do not require asi(X, ρ) be at most1 but instead simply require it to be finite; this produces a dynamic bound ratherthan the customary bound of 2χ(Γ) − 1, and further motivates the question as towhether asi(X, ρ) can be both finite and greater than one.

Theorem 8.1. Let X be a standard Borel space, let Γ a Borel graph on X, and let ρbe the graph metric on Γ. Assume that Γ is locally finite and that s = asi(X, ρ) <∞.Then χB(Γ) ≤ (s+ 1) · (χ(Γ)− 1) + 1.

Proof. Let U0, . . . , Us be Borel sets covering X with F4(Ui) bounded for everyi, and set Vi = B(Ui; 1). Since F2(Vi) is a finite Borel equivalence relation, byProposition 2.4 there is a Borel proper coloring πi : Vi → χ(Γ) of Γ � Vi.

Define c : X → {0}∪ ((s+1)× (χ(Γ)−1)) by setting c(x) = 0 if πi(x) = χ(Γ)−1for every i with x ∈ Vi, and otherwise set c(x) = (i, πi(x)) where i is least withx ∈ Vi and πi(x) 6= χ(Γ)− 1. Then c is Borel. Now suppose (x, y) ∈ Γ. Then thereis i with x ∈ Ui and thus x, y ∈ Vi, which implies c(x) and c(y) cannot both be


0. Say c(x) = (j, p) 6= 0. Then either y 6∈ Vj or else πj(y) 6= p, and in either casec(y) 6= c(x). �

Combining the above theorem with Theorem 4.8.(a) produces the following.

Corollary 8.2. Let X be a standard Borel space, let Γ a locally finite Borel graph onX, and let ρ be the graph metric on Γ. If asdimB(X, ρ) <∞ then χB(Γ) ≤ 2χ(Γ)−1.

Notice that the above corollary provides bounds on the Borel chromatic numbersof Schreier graphs induced by free Borel actions of G whenever G is a finitelygenerated group satisfying the assumption of Theorem 6.4.

Lastly, we briefly discuss the Borel asymptotic dimension of Borel graphs in thehighly restricted setting of acyclic graphs. It is not hard to see that when Γ isacyclic (X, ρ) has standard asymptotic dimension at most 1. However, this canfail in the Borel case. For example, take the graph generated by the free part ofthe shift action of the free group F2 on NF2 , and use [24, Lemma 2.1] to see thatLemma 3.1.(2) fails for every d. Furthermore, in [4] hyperfinite acyclic boundeddegree Borel graphs are constructed that have arbitrarily large Borel chromaticnumber and thus have infinite Borel asymptotic dimension by Corollary 8.2. Onthe other hand, below we show that when a Borel graph is induced by a bounded-to-one Borel function it has Borel asymptotic dimension at most 1. Recall that ifX is a standard Borel space and f : X → X is a Borel function, then Γf is theBorel graph whose vertices are the elements of X and where x, y ∈ X are adjacentif f(x) = y or f(y) = x.

Lemma 8.3. Let X be a standard Borel space, let f : X → X be a bounded-to-one Borel function, and let ρ be the graph metric on Γf . Then (X, ρ) has Borelasymptotic dimension at most 1.

Proof. Fix r > 0. We begin by finding a Borel set Y ⊆ X which has nice recurrenceproperties for f and r. Set A0 = X and f0 = f . Inductively assume that a Borelset An ⊆ X and a bounded-to-one Borel function fn : An → An have been defined.Apply [4, Lem. 4.1] to find a Borel set An+1 ⊆ An such that An+1 shares at mostone point with {x, fn(x)} and at least one point with {fn(x), f2

n(x), f3n(x), f4

n(x)}for every x ∈ An, and for x ∈ An+1 set fn+1(x) = f in(x) where i > 0 is least withf in(x) ∈ An+1. Pick k ∈ N with 2k > 2r and set Y = Ak. Then for some m ∈ N(one can use m = 4k) we have that for every x ∈ X,

|Y ∩ {x, f(x), . . . , f2r(x)}| ≤ 1 ≤ |Y ∩ {f i(x) : 1 ≤ i ≤ m}|.

Let g : X → Y be defined by g(x) = f i(x) where i > 0 is least such thatf i(x) ∈ Y . Let E be the equivalence relation on X where x E y if and only ifg(fr(x)) = g(fr(y)). Then E is uniformly bounded since ρ(x, g(x)) ≤ m for all x.Lastly, we claim that for every x ∈ X the ball B(x; r) meets at most 2 classes ofE. Indeed, fr(B(x; r)) ⊆ {x, f(x), . . . , f2r(x)} and g can take at most 2 values on{x, f(x), . . . , f2r(x)} since |Y ∩ {f(x), . . . , f2r+1(x)}| ≤ 1. �

Lemma 8.3 cannot be generalized to finite-to-one Borel functions. For example,let [N]∞ be the standard Borel space of infinite subsets of N, and let f : [N]∞ → [N]∞

be the shift function where f(A) = A\{min(A)}. Letting ρ be the graph metric onΓf , we have that ([N]∞, ρ) does not have finite Borel asymptotic dimension. Thisfollows from condition (2) of Lemma 3.1 and the Galvin-Prikry theorem [20, Thm.


19.11]. Specifically, if U0, . . . , Ud are Borel sets covering [N]∞, then there is someA ∈ [N]∞ and i ∈ d+ 1 with fn(A) ∈ Ui for all n.

9. Følner tilings

For an action G y X, a finite set K ⊆ G, and δ > 0, we say that a finiteset F ⊆ X is (K, δ)-invariant if |(K · F )4F | < δ|F |. We similarly say that afinite set F ⊆ G is (K, δ)-invariant if |(KF )4F | < δ|F |. Recall from the Følnercharacterization of amenability that a countable group G is amenable if and only iffor every finite set K ⊆ G and δ > 0 there exists a finite (K, δ)-invariant subset of G.The concept of amenability appears as a common theme throughout much of ergodictheory, underlying the classical ergodic theorems, Kolmogorov–Sinai entropy theory,and the foundation of orbit equivalence theory.

A key method for applying amenability in dynamics is the use of Følner tilings(i.e. partitions of the space into finite sets that are (K, δ)-invariant for some pre-scribed K and δ). In ergodic theory, the weaker property of Følner quasi-tilingsoften suffices and was established in the seminal work of Ornstein and Weiss thatextended Kolmogorov–Sinai entropy theory and Ornstein theory to all countableamenable groups [26]. Outside of ergodic theory, quasi-tilings no longer suffice andstronger types of tilings are needed. This issue is receiving renewed attention afterthe recent proof by Downarowicz, Huczek, and Zhang of the purely group-theoreticfact that every countable amenable group can be tiled by Følner sets[8]. This ledto the discovery that free measure-preserving actions of countable amenable groupsalways admit measurable Følner tilings on an invariant conull set [5], and thatcontinuous actions of groups of subexponential volume-growth on 0-dimensionalcompact metric spaces always admit clopen Følner tilings [9].

In this brief section we show that finite Borel asymptotic dimension (or moregenerally, finite asymptotic separation index) is sufficient for the existence of BorelFølner tilings. A similar topological result providing clopen Følner tilings is provenin the next section. Combined with Theorem 6.4 this produces a large class of newgroups whose actions admit Følner tilings.

Theorem 9.1. Let G be a countable amenable group, let X be a standard Borelspace, and let G y X be a free Borel action. Assume that asi(H y X) < ∞ forevery finitely generated subgroup H ≤ G. Then for every finite K ⊆ G and δ > 0there exist (K, δ)-invariant finite sets F1, . . . , Fn ⊆ G and Borel sets C1, . . . , Cn ⊆X such that the map θ :

⊔ni=1 Fi × Ci → X given by θ(f, c) = f · c is a bijection.

Proof. Without loss of generality, we may assume that 1G ∈ K and thus a set F is(K, δ)-invariant if and only if |(K · F ) \ F | < δ|F |. Set H = 〈K〉, d = asi(H y X),and K0 = K. Inductively assume that finite sets K0, . . . ,Ki ⊆ H have been defined.By [8] there are finite (K0 · · ·Ki, δ)-invariant sets Ai0, . . . , A

iì⊆ H, all containing

1G, and sets T i0, . . . , Tiì⊆ H such that {Aipt : p ≤ ì, t ∈ T ip} partitions H. Set

Ki+1 = Ki ∪⋃p≤ì A

ip(A

ip)−1 ⊆ H.

Now set B = K−1d+1Kd+1 ⊆ H and let {U0, . . . , Ud} be a Borel cover of X such

that FB(Ui) is finite for every i. Let Mi ⊆ Ui be a Borel set containing exactly onepoint from every FB(Ui)-class. Define the Borel sets

Zip = {t · x : x ∈Mi, t ∈ T ip, and Aipt · x ∩ [x]FB(Ui) 6= ∅}


and define U+i =

⋃p≤ì A

ip ·Zip. Notice that Ui ⊆ U+

i ⊆ Kd+1 ·Ui and that the sets

{Aip · z : p ≤ ì, z ∈ Zip} partition U+i . Next let Y ip consist of those points z ∈ Zip

for which Aip · z is disjoint with⋃j>i U

+j and set Vi =

⋃p≤ì A

ip · Y ip . Then the Vi’s

are pairwise disjoint and each Vi is partitioned by {Aip · y : p ≤ ì, y ∈ Y ip}Define a Borel function s : X →

⋃i Vi as follows. For x ∈ U+

d = Vd set s(x) = x.

Now inductively assume that s is defined on⋃j>i U

+j and consider x ∈ U+

i . If

x ∈ Vi then set s(x) = x. Otherwise pick p ≤ ì and z ∈ Zip with x ∈ Aip · z and,using any fixed enumeration of G, set s(x) = s(a · z) where a is the least elementof Aip satisfying a · z ∈

⋃j>i U

+j . Then s is Borel, s(x) = x for all x ∈

⋃i Vi, and

when x ∈ U+i and x 6= s(x) ∈ Vj we have j > i and x ∈ Ki+1 · · ·Kj · s(x).

Now define a Borel equivalence relation E by declaring (x, x′) ∈ E if and only ifthere is i with s(x), s(x′) ∈ Vi and there are p ≤ ì and y ∈ Y ip with s(x), s(x′) ∈Aip · y. In other words, the classes of E consist of the sets s−1(Aip · y) for i ∈ d+ 1,

p ≤ ì, and y ∈ Y ip . Since Aip · y ⊆ s−1(Aip · y) ⊆ K1 · · ·KiAip · y, we have

|(K · s−1(Aip · y)) \ s−1(Aip · y)| ≤ |(K0K1 · · ·KiAip) \Aip| < δ|Aip| ≤ δ|s−1(Aip · y)|,

and thus every E-class is (K, δ)-invariant.Notice that [y]E ⊆ K1 · · ·Kd+1·y for every y ∈

⋃i∈d+1

⋃p≤ì Y

ip . Let F0, F1, . . . , Fn

enumerate the (K, δ)-invariant subsets of K1 · · ·Kd+1. For every m ≤ n let Cm bethe set of y ∈

⋃i∈d+1

⋃p≤ì Y

ip satisfying [y]E = Fm · y. Then Cm is Borel since

y ∈ Cm if and only if y ∈⋃i∈d+1

⋃p≤ì Y

ip , (y, f · y) ∈ E for every f ∈ Fm, and

(y, h · y) 6∈ E for every h ∈ (K1 · · ·Kd+1) \ Fm. Lastly, our construction ensuresthat the map θ :

⊔ni=1 Fi × Ci → X given by θ(f, c) = f · c is a bijection. �

10. Topological dynamics and C∗-algebras

In this last section we observe that the proof of our main theorem, Theorem 6.4,easily adapts to the realm of topological dynamics. Moreover, we will see that theproof of Theorem 9.1 adapts as well and that together these results have importantconsequences to C∗-algebras.

We first record a basic lemma.

Lemma 10.1. Let X be a 0-dimensional second countable Hausdorff space and letG y X be a free continuous action. Let U ⊆ X be clopen and let B ⊆ G be finiteand suppose that FB(U) is uniformly finite. Then:

(i) for every g ∈ G the set {x ∈ U ∩ g−1 ·U : (x, g ·x) ∈ FB(U)} is clopen; and(ii) there is a clopen set Y ⊆ U that contains exactly one point from every

FB(U)-class.

Proof. Since FB(U) is uniformly finite, there is a finite set A ⊆ G with [x]FB(U) ⊆A ·x for all x ∈ X (specifically, one can take A = (B∪B−1∪{1G})n for sufficientlylarge n).

(i). Let P be the set of all finite sequences a0, a1, . . . , an ∈ A satisfying a0 = 1G,ai 6= aj for all i 6= j, and satisfying either ai+1 ∈ Bai or ai ∈ Bai+1 for every i < n.Then P is finite and for g ∈ G and x ∈ U ∩g−1 ·U we have (x, g ·x) ∈ FB(U) if and

only if there is (a0, . . . , an) ∈ P with an = g and satisfying x ∈ a−1i · U for every

0 ≤ i ≤ n. Thus {x ∈ U ∩ g−1 · U : (x, g · x) ∈ FB(U)} is clopen.(ii). Since the action is free, for any x ∈ X we have that |A·x| = |A| and therefore

for every open set V containing x there is a clopen set V ′ ⊆ V containing x with


the sets a · V ′, a ∈ A, pairwise disjoint. So there is a base V for the topology on Xsuch that each V ∈ V is clopen and the sets a ·V , a ∈ A, are pairwise disjoint. SinceX is second countable, there is a countable base {Vn : n ∈ N} ⊆ V for the topologyon X. Notice that no two points of Vn ∩ U can be FB(U)-equivalent. Define Ynto be the set of v ∈ Vn ∩ U with [v]FB(U) ∩

⋃i<n Vi = ∅. Then Yn is clopen since

v ∈ Yn if and only if v ∈ Vn ∩ U and for every a ∈ A we either have a · v 6∈ U ,(v, a · v) /∈ FB(U), or a · v 6∈

⋃i<n Vi. Therefore Y =

⋃n Yn is open. Moreover,

Y is also closed since X \ Y is the union of the clopen sets Vn \ (Yn ∪⋃i<n Vi).

Lastly, it is immediate from the definition that Y meets every FB(U)-class preciselyonce. �

In the realm of topological dynamics, the appropriate analog of asymptotic di-mension is the notion of dynamic asymptotic dimension as defined by Guentner,Willet, and Yu in [17]. Specifically, for a discrete group G acting continuously on alocally compact Hausdorff space X, the dynamic asymptotic dimension is d if d ∈ Nis least with the property that for any compact set K ⊆ X and finite set B ⊆ Gthere are open sets U0, . . . , Ud that cover K such that for every i the set TUi

B is finite,

where g ∈ TUi

B if and only if there are x ∈ Ui, m ≥ 1, and b1, . . . , bm ∈ B∪B−1 withg = bmbm−1 · · · b1 and bkbk−1 · · · b1 · x ∈ Ui for every 1 ≤ k ≤ m. When no suchd exists the dynamic asymptotic dimension is defined to be ∞. When the actionis free (as we will always assume), one can instead simply require that FB(Ui) beuniformly finite for every i.

For our purposes, we will consider 0-dimensional spaces and will want the strongerproperty that the sets U0, . . . , Ud are clopen and cover all of X rather than a givencompact subset of X. We first make the easy observation that this stronger condi-tion is automatic when X itself is compact.

Lemma 10.2. Let G be a countable group, X a 0-dimensional compact Hausdorffspace, and G y X a free continuous action. If the action has dynamic asymp-totic dimension d < ∞ then for every finite set B ⊆ G there is a clopen cover{U0, . . . , Ud} of X such that FB(Ui) is uniformly finite for every i.

Proof. Let B ⊆ G be finite. Since X is compact and the dynamic asymptotic di-mension is d, there is an open cover {V0, . . . , Vd} of X such that FB(Vi) is uniformlyfinite for every i. It is easily seen that any disjoint pair of closed subsets of X isseparated by a clopen set. So we can define clopen sets Ui inductively by requiringUi ⊆ Vi to be a clopen set containing X \ (

⋃j<i Uj ∪

⋃j>i Vj). Then the Ui’s are

clopen and cover X, and FB(Ui) is uniformly finite since Ui ⊆ Vi. �

The following lemma is a topological analog of Theorem 4.2. We remark that inthis setting we only obtain a bound on dynamic asymptotic dimension rather thanequality. This is due to the fact that for actions on non-compact spaces, dynamicasymptotic dimension can be less than the standard asymptotic dimension of theacting group [17, Rem. 2.3.(ii)].

Lemma 10.3. Let G be a countable group, X a 0-dimensional second countableHausdorff space, and let G y X be a free continuous action. Let D ∈ N andassume that for every finite set B ⊆ G there is a clopen cover {U0, . . . , UD} ofX such that FB(Ui) is uniformly finite for every i. Then the same is true whenD = asdim(G). In particular, when X is in addition locally compact, the dynamic


asymptotic dimension of the action G y X is at most the standard asymptoticdimension of G.

Proof. We will retrace the proof of Theorem 4.2 and the supporting Lemmas 4.3and 4.4 with clopen sets in place of Borel sets. Fix any proper right-invariant metricτ on G, and let ρ be the extended metric on X satisfying ρ(g ·x, h ·x) = τ(g, h) forall g, h ∈ G and x ∈ X and satisfying ρ(x, y) = ∞ whenever G · x 6= G · y. Noticethat by definition asdim(G) is equal to asdim(G, τ) = asdim(X, ρ).

Let’s say that a clopen set Y ⊆ X has clopen (r,R)-dimension q if there existclopen sets V0, . . . , Vq covering X such that for every i each class of Fr(Ui) hasdiameter at most R. In analogy with Lemma 4.3, we claim that if Y ⊆ X isclopen, Fr(Y ) is uniformly bounded, and (G, τ) has (r,R)-dimension d, then Y hasclopen (r,R) dimension at most d. Indeed, by Lemma 10.1 there is a clopen setY∗ ⊆ Y that meets every Fr(Y )-class in exactly one point. Let {T0, . . . , Td} be anypartition of G with the property that for every i each class of Fr(Ti) has diameterat most R. Let w > 0 bound the diameter of every Fr(Y )-class, define the finitesets T ′i = {t ∈ Ti : τ(1G, t) ≤ w}, and set

Ui =⋃t∈T ′i

{t · y : y ∈ Y∗ ∩ t−1 · Y and (y, t · y) ∈ Fr(Y )}.

Then U0, . . . , Ud are clopen by Lemma 10.1 and they partition Y . Additionally, anyFr(Ui)-class is a subset of a Fr(Ti · y)-class for some y ∈ Y∗ and thus has diameterat most R. So Y has clopen (r,R)-dimension at most d as claimed.

Next, in analogy with Lemma 4.4, we claim that if X is partitioned by the clopensets X0, . . . , Xn−1 and each Xi has clopen (15ri, ri+1)-dimension at most d, wherethe rn’s are non-decreasing, then (X, ρ) has clopen (r0, 5rn)-dimension at most d.Indeed, one can follow the proof of Lemma 4.4 and change each instance of “Borel”to “clopen.”

Finally, we retrace the proof of Theorem 4.2. Let D be as in the statement ofthis lemma, set d = asdim(G), and let B ⊆ G be finite. Since asdim(X, ρ) = d wecan find an increasing sequence of radii ri such that r0 > max{τ(b, 1G) : b ∈ B} andsuch that X has standard (15ri, ri+1)-dimension at most d for every i. From ourassumption we can obtain a clopen partition {X0, . . . , XD} of X such that FrD (Xi)is uniformly bounded for every i. Our analog of Lemma 4.3 implies that each Xi

has clopen (15ri, ri+1)-dimension at most d, and thus our analog of Lemma 4.4implies that (X, ρ) has clopen (r0, 5rD+1)-dimension at most d. This means thatthere exists a clopen cover {U0, . . . , Ud} of X such that for every i each class ofFr0(Ui) has diameter at most 5rD+1. From our definition of r0 we then see thatFB(Ui) is uniformly finite for every i. �

We next turn to proving a topological version of Theorem 6.4. We will proceedmuch in the same way as before.

Definition 10.4. For countable groups G � H, lets say that (G,H) has goodclopen asymptotics if for every 0-dimensional second countable Hausdorff space Xand every free continuous action H y X the following two properties hold:

(i) there is d ∈ N so that for every finite set B ⊆ G there is a clopen cover{U0, . . . , Ud} of X such that FB(Ui) is uniformly finite for every i; and


(ii) for every finite set C ⊆ H there is ` ∈ N so that for every finite set B ⊆ Gthere is a clopen cover {W0, . . . ,W`} of X consisting of CB\G-independentsets.

Lemma 10.5. Let F � H be countable groups with (F,H) having good clopenasymptotics. Let X be a 0-dimensional second countable Hausdorff space, let H yX be a free continuous action, and let d <∞ be as in Definition 10.4.(i). Then forany finite sets C ⊆ H and A ⊆ F there is a clopen cover {Uni : i ∈ d+ 1, n ∈ `+ 1}of X such that Uni is CA\F -independent and FA(Uni ) is uniformly finite, and thereare finite sets CA∩ F ⊆ An ⊆ F such that whenever Z is a FAn

(Uni )-class, c ∈ C,and m > n, the set cA · Z is either disjoint with Umi or else contained in a singleFAm

(Umi )-class.

Proof. We argue that the proof of Lemma 6.2 can be followed but with clopen setsin place of Borel sets. Specifically, in that proof we can choose the sets V ni andWn to be clopen rather than Borel. Notice that the explanation given there forwhy sni (X ′) is Borel when X ′ is Borel shows, when combined with Lemma 10.1,that sni (X ′) is clopen when X ′ is clopen. Therefore the sets U0

i = V 0i ∩W0 and

Uni = s0i ◦ · · · ◦ s

n−1i (V ni ∩Wn), n ≥ 1, are clopen. �

Lemma 10.6. Let H be a countable group and let F � G be normal subgroups ofH. Assume that (F,H) has good clopen asymptotics and that G/F has {φc : c ∈C}-bounded packing whenever C ⊆ H is a finite set with the property that eachautomorphism φc(gF ) = cgc−1F is either trivial or an outer automorphism. Then(G,H) has good clopen asymptotics.

Proof. Let X be a 0-dimensional second countable Hausdorff space and let H y Xbe a free continuous action. Let dF be as in Definition 10.4.(i) for (F,H). Wefollow the proof of Lemma 6.3. By replacing Lemma 6.2 with Lemma 10.5, wecan choose the sets Uni to be clopen. It is immediate that the sets V `Fi = U `Fiand V ni = Uni \

⋃m>nB · V mi are clopen. Since the value πi(x), for x ∈ V ni ,

is defined to be 0 if n = `F and for n < `F is defined to be the least element of(kG/F+1)\{πi(t·x) : t ∈ B3CB3 and t·x ∈

⋃m>n V

mi }, it follows immediately from

induction that the function πi is continuous. Therefore the sets W ji = B · π−1

i (j)are clopen. The concluding arguments of the proof of Lemma 6.3 then show that(G,H) has good clopen asymptotics. �

Theorem 10.7. Let G be a countable group admitting a normal series where eachquotient of consecutive terms is a finite group or a torsion-free abelian group withfinite Q-rank, except the top quotient can be any group of uniform local polynomialvolume-growth or the lamplighter group Z2 o Z. If X is a 0-dimensional secondcountable Hausdorff space, G y X is a continuous free action, and B ⊆ G isfinite, then there is clopen cover {U0, . . . , Ud} of X where d = asdim(G) < ∞ andFB(Ui) is uniformly finite for every i. In particular, all free continuous action of Gon locally compact 0-dimensional second countable Hausdorff spaces have dynamicasymptotic dimension at most asdim(G) <∞.

Proof. Follow the proof of Theorem 6.4, but with Lemma 10.6 replacing Lemma 6.3and Lemma 10.3 replacing Theorem 4.2. For verifying the base case of the induction,recall from the proof of Lemma 10.1 that, given a finite set C ⊆ G, there is a base{Vn : n ∈ N} for the topology on X such that each Vn is clopen and c ·Vn ∩Vn = ∅


for all c ∈ C \ {1G}. Inductively define a continuous function π : X → |C| + 1 byletting π(x) be the least element of (|C| + 1) \ {π(c · x) : c ∈ C, c · x ∈

⋃k<n Vk}

for x ∈ Vn \⋃k<n Vk. Then {π−1(i) : i ∈ |C|+ 1} is a clopen cover of X consisting

of C \ {1G}-independent sets. �

The theorem below together with Lemma 10.2 and Theorem 9.1 completes theproof of Theorem 1.4 from the introduction.

Theorem 10.8. Let G be a countable amenable group, let X be a 0-dimensionalsecond countable Hausdorff space, and let G y X be a free continuous action.Assume that for every finitely generated subgroup H ≤ G there is d ∈ N so that forevery finite set B ⊆ H there is a clopen cover {U0, . . . , Ud} of X such that FB(Ui)is uniformly finite for every i. Then for every finite K ⊆ G and δ > 0 there exist(K, δ)-invariant finite sets F1, . . . , Fn ⊆ G and clopen sets C1, . . . , Cn ⊆ X suchthat the map θ :

⊔ni=1 Fi × Ci → X given by θ(f, c) = f · c is a bijection.

Proof. We follow the proof of Theorem 9.1 but use clopen sets. As before let H =〈K〉 but now let d be as described in our assumption for H. We can then choose thesets U0, . . . , Ud to be clopen and with the property that FB(Ui) is uniformly finitefor every i. Consequently, there is a finite set Q ⊆ G with Kd+1 · [x]FB(Ui) ⊆ Q · xfor every x ∈ Ui (one can use Q = (B∪B−1∪{1G})n for sufficiently large n). Nextwe use Lemma 10.1 to choose the Mi’s to be clopen. Since T pi can be replacedwith the finite set T pi ∩Q in the definition of Zpi , we see that each Zpi is clopen byLemma 10.1. Similarly, U+

i , Y pi , and Vi are clopen.Next, we claim that the function t : X → H defined by the condition s(x) =

t(x) · x is continuous. For x ∈ U+d = Vd we have s(x) = x and thus t(x) = 1G. So

t is continuous on U+d . Now inductively assume that t is continuous on

⋃j>i U

+j .

For x ∈ Vi we have s(x) = x and t(x) = 1G. When x ∈ U+i \ Vi we pick p ≤ ì and

a1 ∈ Aip satisfying a−11 · x ∈ Zpi , we pick the least a2 ∈ Aip satisfying a2a

−11 · x ∈⋃

j>i U+j , and set s(x) = s(a2a

−11 · x), meaning t(x) = t(a2a

−11 · x)a2a

−11 . Since i,

p, a1, and a2 depend continuously on x, we conclude that t is continuous.It only remains to check that each set Cm constructed as in the proof of Theorem

9.1 is clopen. This is the case because y ∈ Cm if and only if there is i ∈ d+ 1 andp ≤ ì with y ∈ Y ip and with the property that t(f · y) ∈ Aipf−1 for every f ∈ Fmand t(h · y) 6∈ Aiph−1 for all h ∈ (K1 · · ·Kd+1) \ Fm. �

The combination of Theorems 10.7 and 10.8 allow us to find new examples ofactions that are almost finite. From this property we derive consequences to C∗-algebras, ultimately finding new examples of classifiable crossed products.

Corollary 10.9. Let G be a countable group that is a union of groups satisfying theassumption of Theorem 10.7. Let X be a compact metrizable space and let Gy Xbe a free continuous action. Assume either that X has finite covering dimension orthat the action Gy X has the topological small boundary property. Then:

(i) the action Gy X is almost finite;(ii) if additionally Gy X is minimal, then C(X)oG is Z-stable and has finite

nuclear dimension.

Proof. Statement (i) follows from Theorem 10.7, Theorem 10.8, and [22, Thm. 10.2]when X is 0-dimensional, and from [23, Thm. 7.6 and Cor. 7.7] in the remaining


cases. When the action is minimal, Z-stability follows from (i) and [22, Thm. 12.4],and finite nuclear dimension follows from Z-stability and [3, Thm. A]. �

Corollary 10.10. Consider the collection of free minimal continuous actions GyX where G is a countable group that is a union of groups satisfying the assumptionof Theorem 10.7, X is a compact metrizable space, and either X has finite coveringdimension or the action Gy X has the topological small boundary property. Thenthe crossed products arising from these actions are classified by the Elliot invari-ant (ordered K-theory paired with tracial states) and are simple ASH algebras oftopological dimension at most 2.

Proof. The crossed products C(X)oG of these actions have finite nuclear dimensionby Corollary 10.9, satisfy the universal coefficient theorem (since the groups weconsider are amenable) [30, Prop. 10.7], and are stably finite (since they come fromfree minimal actions of amenable groups). Therefore these crossed products areclassified by their Elliot invariants [29, Cor. D] and are simple ASH algebras oftopological dimension at most 2 [29, Thm. 6.2.(iii)]. �

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BOREL ASYMPTOTIC DIMENSION AND HYPERFINITE …bseward/Files/asymdim.pdfymptotic dimension and its applications to group theory, see [2]). In recent years, a related notion of dynamic